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MATHEMATICAL    MONOGRAPHS. 

EDITED    BV 

MANSFIELD  MERRIMAN  and  ROBERT   S.    WOODWARD. 


No.  11 
FUNCTIONS 

OF  A 

COMPLEX    VARIABLE 


BY 

THOMAS   S.   FISKE, 

Professor  of  Mathematics  in  Columbia  Univbrsitv* 


FOURTH    EDITION. 
FIRST    THOUSAND. 


NEW   YORK: 

JOHN    WILEY    &    SONS. 

London:    CHAPMAN    &    HALL,    Limited. 
1907 


Goi 


ComuGHT,  1896, 

BY 

MANSFIELD  MERRIMAN  and  ROBERT  S.  WOODWARD 

UNDER  THE   TITLE 

HIGHER    MATHEMATICS. 

First  Edition,  September,  1896. 
Second  Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,   November,  1906. 


ItOnitltT  DRUMMOND,   PRINTER,   NEW  VORK* 


EDiTORS'  PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  ampUfy  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  hterature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elliptic  functions,  the  theory  of  num- 
bers, the  group  theory,  the  calculus  of  variations,  and  non- 
Euclidean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  pubHcation  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


AUTHOR'S  PREFACE. 


In  the  following  pages  is  contained  a  brief  introductory 
account  of  some  of  the  more  fundamental  portions  of  the  theory 
of  functions  of  a  complex  variable.  The  work  was  prepared 
originally  as  a  chapter  for  the  volume  called  "  Higher  Mathe- 
matics," pubUshed  in  1896.  It  has  been  enlarged  by  the  addition 
of  sections  on  power  series,  algebraic  functions  and  their  integrals, 
functions  of  two  or  more  independent  variables,  and  differential 
equations.  Furthermore,  the  section  on  uniform  convergence 
has  been  extended,  and  the  treatment  of  Weierstrass's  theorem 
and  of  Mittag-Leffler's  theorem  has  been  simplified. 

It  is  hoped  that  the  present  work  will  give  the  uninitiatea 
some  idea  of  the  nature  of  one  of  the  most  important  branches 
of  modem  mathematics,  and  will  also  be  useful  as  an  introduction 
to  larger  works,  such  as  those  in  English  by  Forsyth,  Whittaker, 
and  Harkness  and  Morley;  in  French  by  Jordan,  Picard,  Goursat, 
and  Valine- Poussin;  and  in  German  by  Burkhardt,  Stolz  and 
Gmeiner,  and  Osgood. 

New  York,  August,  1906. 


CONTENTS. 


Art.    I.  Definition  of  Function •     •     .     .    Page  i 

2.  Representation  of  Complex  Variable      .     • 2 

3.  Absolute  Convergence ,.  3 

4.  Elementary  Functions 4 

5.  Continuity  of  Functions 5 

6.  Graphical  Representation  of  Functions 7 

7.  Derivatives 8 

8.  conformal  representation ii 

9.  Examples  of  Conformal  Representation 13 

10.    Conformal  Representation  of  a  Sphere .  19 

■  II.    Conjugate  Functions io 

12.  Application  to  Fluid  Motion 21 

13.  Singular  Points 25 

14.  Point  at  Infinity 31 

15.  Integral  of  a  Function 32 

16.  Reduction  of  Complex  Integrals  to  Real 36 

17.  Cauchy's  Theorem 37 

18.  Application  of  Cauchy's  Theorem       ,     .     .' 39 

19.  Theorems  on  Curvilinear  Integrals 42 

20.  Taylor's  Series 44 

21.  Laurent's  Series 46 

22.  Fourier's  Series 48 

2,3.    Uniform  Convergence 46 

24.  Power  Series 54 

25.  Uniform  Convergence  of  Pov^^er  Series 56 

26.  Uniform  Functions  with  Singular  Points 57 

27.  Residues 61 

28.  Integral  of  a  Uniform  Function 63 

29.  Weierstrass's  Theorem 66 

30.  Mittag-Leffler's  Theorem 71 

31.  Singular  Lines  and  Regions 78 

32.  Functions  having  n  Values ,.  81 

33.  Algebraic  Functions 83 

34.  Integrals  of  Algebraic  Functions 85 

35.  Functions  of  Several  Variables .  89 

36.  Differential  Equations 91 

IndSx 99 

V 


FUNCTIONS  OF  A  COMPLEX  VARIABLE. 


Art.  1.    Definition  of  Function. 

If  two  or  more  quantities  are  such  that  no  one  of  them  suf- 
fers any  restriction  in  regard  to  the  values  which  it  can  assume 
when  any  values  whatsoever  are  assigned  to  the  others,  the 
quantities  are  said  to  be  ♦*  independent." 

A  quantity  is  said  to  be  a  function  of  another  quantity  or  of 
several  independent  quantities  if  the  former  is  determined  in 
value  whenever  particular  values  are  assigned  to  the  latter. 
The  quantity  or  quantities  upon  the  values  of  which  the  value 
of  the  function  depends,  are  said  to  be  the  '*  independent  vari- 
ables "  of  the  function. 

A  function  is  *'  one-valued,"  or  *' uniform,"  when  to  every 
set  of  values  assigned  to  the  independent  variables  there  cor- 
responds but  one  value  of  the  function.  It  is  said  to  be 
**w-valued"  when  to  every  set  of  values  of  the  independent 
variables  n  values  of  the  function  correspond. 

The  ''Theory  of  Functions"  has  among  its  objects  the 
study  of  the  properties  of  functions,  their  classification  accord- 
ing to  their  properties,  the  derivation  of  formulas  which  exhibit 
the  relations  of  functions  to  one  another  or  to  their  independ- 
ent variables,  and  the  determination  whether  or  not  functions 
exist  satisfying  assigned  conditions. 


•  Ft/jfcf i(5n8  of  a  complex  vakiablb. 


Art.  2.    Representation  of  Complex  Variable. 

A  variable  quantity  is  capable,  in  general,  of  assuming  both 
real  and  imaginary  values.  In  fact,  unless  it  be  otherwise 
specified,  every  quantity  w  is  to  be  regarded  as  having  the 
"  complex  "  form  u-\-v    V—  i,  u  and  v  being  real.     It  is  cus- 


tomary to  denote  r  —  i  by  t,  and  to  write  the  preceding  quan- 
tity thus  :  M  -}-  ^^'  I^  ^  is  zero,  w  is  real ;  if  u  is  zero,  zv  is  a 
**  pure  imaginary." 

A  quantity  2  =  x-\-  ty  is  said  to  vary" continuously  "  when 
between  every  pair  of  values  which  it  may  take,  c^  =  a^-\-  ib^^ 
^2  =  ^2  4"  ^^2 »  ^  and_;/  must  pass  through  all  real  values  inter- 
mediate to  a^  and  a^ ,  b^  and  b^ ,  respectively,  either  once  or  a 
'inite  number  of  times. 

It  is  usual  to  give  to  a  variable  quantity  ^  =  ;r  +  ^^  a  graphi- 
cal representation  by  drawing  in  a  plane  a  pair  of  rectangular 
axes  and  constructing  a  point  whose  abscissa  and  ordinate  are 
respectively  equal  to  x  and  y.  To  every  value  of  z  will  corre- 
spond a  point ;  and,  conversely,  to  every  point  will  correspond 
a  value  of  z.  The  terms  '*  point  "  and  value,  then,  may  be  inter- 
ohanged  without  confusion.  When  z  varies  continuously  the 
graphical  representation  of  its  varia- 
tion, or  its  *'  path,"  will  be  a  continuous 
line.  This  graphical  representation  is 
of  the  highest  importance.  By  means 
of  it  some  of  the  most  complicated 
propositions  may  be  given  an  exceed- 
ingly condensed  and  concrete  expres- 
sion. 

By  putting  x  =  r  cos  6,  y  =  r  sin  6,  where  r  is  a  positive  real 
quantity,  the  point 

z  =  r(cos  6  -{-t  sin  6) 

is  referred  to  polar  coordinates.  The  quantity  r  is  called  the 
absolute  value  or  "  modulus  "  of  z.  It  will  often  be  written  \js\. 
0  is  known  as  the  ^argument  "  of  2. 


ABSOLUTE   CONVERGENCE.  3 

A  function  Is  sometimes  considered  for  only  such  values  of 
each  independent  variable  as  are  represented  graphically  by  the 
points  of  a  certain  continuous  line.  In  the  study  of  functions 
of  real  variables,  for  example,  the  path  of  each  independent 
variable  is  represented  by  a  straight  line,  namely,  the  axis  of 
real  quantities,  or^;'  =  o. 

*Art.  3.    Absolute  Convergence. 

The  representation  of  functions  by  means  of  infinite  series 
is  one  of  the  most  important  branches  of  the  theory  of  func- 
tions. In  many  problems,  in  fact,  it  is  only  by  means  of  series 
that  it  is  possible  to  determine  functions  satisfying  the  condi- 
tions assigned  and  to  obtain  the  required  numerical  results. 
Frequent  use  will  be  made  of  the  following  theorem. 

Theorern. — If  the  moduU  of  the  terms  of  a  series  form  a 
convergent  series,  the  given  series  is  convergent. 

Let  the  given  series  be  W  =  w^-\-w^-\- .  .  .  +  ^n  +  •  •  • 
in  which  zv^  =  r^  (cos  b^  +  e  sin  B^,  w,  =  r,  (cos  6^  +  zsin  6^,) .  .  . 
By  hypothesis  the  series  R=  r^-\-r^-\-  .  .  .  +  r„  +  .  .  .  is 
convergent.  Its  terms  being  all  positive,  the  sum  of  its  first  m 
terms  constantly  increases  with  m,  but  in  such  a  manner  as  to 
approach  a  limit.  The  same  will  be  true  necessarily  of  any 
series  formed  by  selecting  terms  from  R.  The  sum  of  the  first 
m  terms  of  the  series  W  is  composed  of  two  parts, 

r,  cos  e^  +  r,cose^    .  .  .  +  r^-.  cos  6^,,, 
i{r,  sin  0^  +  r,  sin  ^,  + .  .  .  +  r^_,  sin  0^.,), 

and  each  of  these  in  turn  may  be  divided  into  parts  which  have 
all  their  terms  of  the  same  sign.  Every  one  of  the  four  parts 
thus  obtained  approaches  a  limit  as  m  is  increased  ;  for  the 
terms  of  each  part  have  the  same  sign,  and  cannot  exceed^  in 
absolute  value,  the  corresponding  terms  of  R,  Hence,  as  m  is 
increased,  the  sum  of  the  first  m  terms  of  W  approaches  a 
limit;  which  was  to  be  proved. 

A  series,  the  moduli  of  whose  terms  form  a  convergent 
series,  is  said  to  be  "  absolutely  convergent." 


4  FUNCTIONS   OF  A   COMPLEX   VARIABLE. 

Prob.  I.  Show  that  the  series  i  -}-  2;  -f-  ^'  +  •  •  •  +  ^**  +  •  •  'is 
absolutely  convergent,  if   |  ^  |  <  i. 

Art.  4.    Elementary  Functions. 

In  elementary  mathematics  the  functions  are  usually  con- 
sidered for  only  real  values  of  the  independent  variables.  In 
the  case  of  the  algebraic  functions,  however,  there  is  no  diffi- 
culty in  assuming  that  the  independent  variables  are  complex. 
The  theory  of  elimination  shows  that  every  algebraic  equation 
can  be  freed  from  radicals.  Every  algebraic  function,  there- 
fore, is  defined  by  an  equation  which  may  be  put  in  a  form 
wherein  the  second  rnember  is  zero  and  the  first  member  is 
rational  and  entire  in  the  function  and  its  independent  variables. 

Besides  the  algebraic  functions,  the  functions  most  often 
occurring  in  elementary  mathematics  are  the  trigonometric  and 
exponential  functions  and  the  functions  inverse  to  them.  The 
definitions,  by  which  these  functions  are  generally  first  intro- 
duced, have  no  significance  in  the  case  where  the  inde- 
pendent variables  are  complex.  However,  the  following 
familiar  series, 

^  =  exp^=  1+^+1  +  !^  +  ^+..., 
2       31      41 

cos^=i--+^-^+..., 

z*        z^        z'    . 
smz=z--  +  -^--^  +  ... 

which  have  been  established  for  the  case  where  the  variables 
are  real,  furnish  most  convenient  general  definitions  for  exp  Zj 
cos  z,  and  sin  z,  these  series  being  absolutely  convergent  for 
every  finite  value  of  z.  Defining  the  logarithmic  function  by 
the  equation__ 

g\ogx  —  exp  (log<3:)  =  Zy 
it  follows  that 

^«  _  ^  loga  _  gxp {z  log  a). 


CONTINUITY   OF   FUNCTIONS, 


The  following  equations  also  are  to  be  regarded  as  equations 
of  definition : 


sin^ 
tan  2  — 

COS-S: 
cot  2  = , 

sin^ 

cos  2 

I 
sec  2  = . 

I 
cosec  2  =  — , 

cos-s  sin  ^ 

It  may  be  shown  that  the  formulas  which  are  usually  obtained 
on  the  supposition  that  the  independent  variables  are  real,  and 
which  express  in  that  case  properties  of  and  relations  between 
the  preceding  functions,  still  hold  when  the  independent 
variables  are  complex. 

Prob.  2.  Show  that  e'^e'*  =  e'*"^  ",  m  and  n  being  complex. 

Prob.  3.   Deduce   cos  z  =  i{e''  +  e-%  sin  z  —  --.(e''  —  e'"). 

Prob.  4.  Deduce  cos  {z^  +  z^  =  cos  z^  cos  z^  —  sin  z^  sin  z^ , 
sin  {z^  4"  ^3)  =  cos  z^  sin  z^  +  sin  5,  cos  z^ , 

Art.  5.    Continuity  of  Functions. 

Let  a  function  of  a  single  independent  variable  have  a 
determinate  value  for  a  given  value  c  of  the  independent  vari- 
able. If,  when  the  independent  variable  is  made  to  approach 
c,  whatever  supposition  be  made  as  to  the  method  of  approach, 
the  function  approaches  as  a  limit  its  determinate  value  at  c, 
the  function  is  said  to  be  "  continuous  "  at  c. 

This  definition  may  be  otherwise  expressed  as  follows :  A 
function  of  a  single  independent  variable  is  continuous  at  the 
point  c,  when,  being  given  any  positive  quantity  e,  it  is  possible 
to  construct  a  circle,  with  center  at  c  and  radius  equal  to  a 
determinate  quantity  d,  so  small  that  the  modulus  of  the 
difference  between  the  value  of  the  function  at  the  center  and 
that  at  every  other  point  within  the  circle  is  less  than  e. 

A  function  of  several  independent  variables  is  said  to  be 
continuous  for  a  particular  set  of  values  assigned  to  those  vari- 
ables, when  it  takes  for  that  set  of  values  a  determinate  value 
c,  and  for  every  new  set  of  values,  obtained  by  altering  the 


0  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

variables  by  quantities  of  moduli  less  than  some  determinate 
positive  quantity  6,  the  value  of  the  function  is  altered  by  a 
quantity  of  modulus  less  than  any  previously  chosen  arbitrarily 
small  positive  quantity  e. 

A  function  of  one  independent  variable  is  said  to  be  con- 
tinuous in  a  given  region  of  the  plane  upon  which  its  indepen- 
dent variable  is  represented,  if  it  is  continuous  at  every  point 
in  that  region. 

From  the  principles  of  limits,  it  follows  that  if  two  functions 
are  continuous  at  a  given  point,  their  sum,  difference,  and  prod- 
uct  are  continuous  at  that  point.  As  an  immediate  conse- 
quence, every  rational  entire  function  of  ^  is  continuous  at 
every  finite  point ;  for  every  such  function  can  be  constructed 
from  2  and  constant  quantities  by  a  finite  number  of  additions, 
subtractions,  and  multiplications. 

Let  a  function  of  a  single  independent  variable  be  contin- 
uous at  c,  and  let  it  take  at  that  point  the  value  /,  different 
from  zero.  Suppose  also  that  at  any  other  point  ^+^^  the 
function  takes  the  value  /  +  ^^-     Then 

I  I  J/ 


t  +  Jt      t  t{t-\-  At) 

If  it  be  assumed  that  |  J/ 1  <  |  /  |,  the  modulus  of  the  preceding 
difference  cannot  exceed 

\At\ 


\t\{\t\-\At\)' 
and  will,  therefore,  be  less  than  e  if 

eUl' 


\At\< 


\+e\t\ 


Hence  if  a  function  is  continuous  and  different  from  zero 
at  a  point  Cy  its  reciprocal  is  also  continuous  at  c.  It  follows 
at  once  that  if  two  functions  are  both  continuous  at  c,  their 
ratio  is  continuous  at  c,  unless  the  denominator  reduces  to  zero 


GRAPHICAL   REPBESEKTATIOIS"  OP  PU2fCTI0IfS.  7 

at  that  point.  But  every  rational  function  of  z  may  be  expressed 
as  the  ratio  of  two  entire  functions.  It  is  therefore  continuous 
for  all  values  of  z  except  those  for  which  its  denominator 
vanishes. 

Consider  the  function  exp^, 

Hence  if  \Az\<\^ 

but  the  limit  of  the  third  member  is  zero  when  \Az\  ap- 
proaches zero.  Hence  exp  z  is  continuous  for  all  finite  values 
of  z, 

Prob.  5.  Show  that  cos  z  and  sin  z  are  continuous  for  all  finite 
values  of  z. 

Prob.  6.  Show  that  tan  z  is  continuous  in  any  circle  described 
about  the  origin  as  a  center  with  a  radius  less  than  ^n. 

Art.  6.    Graphical  Re^presentation  of  Functions. 

It  was  shown  in  Art.  2  that  a  plane  suffices  for  the  complete 
graphical  representation  of  the  values  of  an  independent  vari- 
able. In  the  same  way  it  is  convenient  to  use  a  second  plane 
to  represent  graphically  the  values  of  any  one-valued  function. 
For  example,  if  w  ^=f{z)  be  such  a  function,  to  each  point 
X  -\-iy  o{  the  independent  variable  will  correspond  a  point 
II  +  iv  of  the  function.  This  point  u  +  iv  is  called  the  "  image  " 
of  the  point  x  -\-iy.  If  te^  is  a  continuous  function  of  z,  then 
every  continuous  curve  in  the  <^-plane  will  have  an  image  in 
the  w-plane,  and  this  image  will  be  also  a  continuous  curve. 

Consider  the  expression  u  -^  iv  =  x"^  -\-  y^  -\-  lixy.      Here 


8 


FUNCTIONS   OF   A   COMPLEX   VARIABLE. 


u  =  X*  -\-  y  and  v  =  2xy,  Since  to  every  value  of  2  corre- 
spond determinate  values  of  x  and  j/, 
and  consequently  determinate  values 
of  u  and  v,  this  expression  falls  un- 
der the  general  definition  of  a  func- 
tion of  ^.  It  is  evidently  continuous. 
Every  straight  line  x  =  t  parallel  to 
the  axis  of  /  is  converted  by  means 
of  it  into  a  parabola  v^  =  4t\u  —  /'). 

Prob.  7.  Find  the  family  of  curves 
into  which  the  straight  lines  parallel  to 
the  axis  of  y  are  converted  by  means  of 
the  function  u  -}-  w  =  x^  —  y'  -{-  2ixy. 
of  this  family  imersect. 


Show  that  no  two  curves 


Art.  7.    Derivatives. 

Let  w  =  f{z)  be  a  given  function  of  2.  If  k  is  an  ''  infini- 
tesimal," that  is,  a  variable  having  zero  as  its  limit,  and  if  the 
expression 

'       ■  f[z  +  h)-f{z) 

h 

has  a  finite  determinate  limit,  remaining  the  same  under  all 
possible  suppositions  as  to  the  way  in  which /^  approaches  zero, 
this  limit  is  said  to  be  the  "  derivative"  of  the  function y(^)  at 
the  point  z.  In  this  case  w  =  f{z)  is  said  to  be  "  monogenic  " 
at  z.  The  derivative  is  written  f'{z)  or  -r-.  A  function  is  said 
to  be  monogenic  in  a  region  of  the  plane  of  the  independent 
variable  if  it  is  monogenic  at  every  point  of  that  region. 

Consider  now  the  circumstances  under  which  a  function 
w  :=.  u  -\-  iv  may  have  a  derivative  at  the  point  z  ^^  x  -\-  iy. 
If  2:  be  given  a  real  increment,  x  is  changed  into  x  ■\-  Axy  while 
y  is  unaltered,  so  that  Az  =  Ax\  and 

Aw   __  Au        .  Av 
Az    ~~  Ax     ""  ~Ax' 


DERIVATIVES.  9 

If,  on  the  other  hand,  z  is  given  a  purely  imaginary  incre- 
ment, Az  =  iAyy  and 

Aw  __  Au    _,    Av 

Az   ~~  iAy        Ay  ' 

If  the  second  members  of  these  equations  approach  deter- 
minate limits  as  Ax  and  Ay  approach  zero,  and  if  these  limits 
are  equal, 

d^'^^d^~~"^dy^^y 

Hence,  equating  real  and  imaginary  parts, 

du  _  9^  dv  _      'du 

dx~'dy'         dx~~dy* 

which  are  necessary  conditions  for  the  existence  of  a  derivative. 

It  can  be  shown  that  these  conditions  are  also  sufficient  * 
For  let  the  increment  of  the  independent  variable  be  entirely 
arbitrary,  no  supposition  being  made  as  to  the  relative  magni- 
tudes of  its  real  and  imaginary  parts.  Then  the  diffeiiential  of 
the  function,  that  is,  that  part  of  the  increment  of  the  function 
which  remains  after  subtracting  the  terms  of  order  higher  than 
the  first,  is 

^  \dJtr  ^  dxl  .      '    \dy    '    dy  J  "^ 

Hence  (^u   x-'dA.   l^u^    ,    .^\  dy^ 

-;  .         du  +  idv  _  Va-y        a-y/  "^  W       ^  dyi  dx 
-   '         dx-\-idy~  J    ,dy_ 

•^  dx 

which,  by  virtue  of  the  conditions  written  above,  is  equal  to 
either  member  of  the  equation 

dx  '^  ^dx  ~      ^dy        dy 
The  value  thus  obtained  is  independent  of  -^,  or,  what  is  the 

*  For  a  complete  discirssion  see  article  by  E.  Goursat  in  the  Transactions  of 
the  Amer.  Math.  See,  vol.  i,  p.  14. 


10  FUNCTIONS  OF  A   COMPLEX  VARIABLE. 

same  thing,  of  the  direction  of  approach  to  the  point  z.     The 
existence  of  a  derivative  of  the  function  w  depends,  therefore, 

only  on   the  existence  of  partial  derivatives  ;— -,  ;:— ,  — -,  -- 

^x    ox   djy    ^y 

satisfying  the  specified  equations  of  condition. 

The  same  equations  of  condition  express  the  tact  that 
w  =  «  +  iv^  supposed  to  be  an  analytical  expression  involving 
X  and  7,  and  having  partial  derivatives  with  respect  to  each, 
involves  ^  as  a  whole,  that  is,  may  be  constructed  from  z  by 
some  series  of  operations,  not  introducing  x  ox  y  except  in  the 
combination  x  -\-  iy.  In  other  words,  they  indicate  that  x  and 
y  may  both  be  eliminated  from  w  =  (p(x,  y)  by  means  of  the 
equation  z  =  x  -\-  iy.  This  property,  however,  is  not  sufficient 
to  define  a  function  as  monogenic,  for  not  every  function  which 
possesses  it  has  a  derivative  with  respect  to  z. 

A  monogenic  function  is  necessarily  continuous ;  that  is, 
the  existence  of  a  derivative  involves  continuity.     For,  if 

it  follows  that 

where  tf  approaches  zero  with  h.  Hence  f{z)  is  the  limit  of 
f{z-\-  h)  when  h  approaches  zero,  or  f(z)  is  continuous  at  the 
point  z. 

The  following  pages  relate  almost  exclusively  to  functions 
which  are  monogenic  except  for  special  isolated  values  of  z. 
Functions  which  are  discontinuous  for  every  value  of  the  inde- 
pendent variable,  and  functions  which  are  continuous  but  admit 
no  derivatives,  have  been  little  studied  except  in  the  case  of 
real  variables  * 

*  In  this  connection  see  G.  Darboux,  Sur  les  fonctions  discontinues,  Annales  de 
I'Ecole  Normale,  Series  2,  vol.  4  (1875),  pp.  51-112.  For  a  systematic  treatment 
of  functions  of  a  real  variable,  see  the  German  translation  of  Dini's  treatise  by 
Liiroth  and  Schepp,  Leipzig,  1892.  For  an  illustration  of  a  function  constructed 
from  2  by  a  series  of  arithmetical  operations  and  discontinuous  for  a  particular 
value  of  z,  see  the  expression  given  on  page  53. 


CONFORMAL    REPKESE^^TATIOJ?". 


11 


Art.  8.    Conformal  Representation. 

Let  z  start  from  the  point  z^  and  trace  two  different  paths 
forming  a  given  angle  at  the  point  z^y  and  let  z^  and  z^  be  arbi- 
trary points  on  the  first  and  second  paths  respectively.     Then 

z^—  z^=  r,(cos  6^,  +  ^  sin  d^  =  r/^\ 

where  r,  denotes  the  length  of  the  straight  line  joining  z^  and 


z^ ,  and  6*,  denotes  the  inclination  of  this  line  to  the  axis  of 
reals.     In  the  same  way,  for  the  point  z^y  there  is  an  equation 

z^  —  z^=  r, (cos  8^  +  ^  sin  6*,)  =  r/^-^. 

If  now  w  is  a  one-valued  monogenic  function  of  z,  in  the 
region  of  the  <s:-plane  considered,  to  the  points  z^,  z^,  z^  corre- 
spond points  w^yW^yW^'^  and  for  these  points  can  be  formed 
the  equations 

^1  —  ^0  =  Pi^'** »    w^  —  w^—  p,^'*'. 

From  the  supposition  that  w  is  monogenic,  it  follows  at 
once  that,  when  z^  and  z^  are  assumed  to  approach  z^^ 

hmit  ~ °  =  limit  — ^ ?. 

z,  —  z,  z,-  z. 

If  the  members  of  this  equation  are  not  equal  to  zero,  it  may 
be  put  in  the  form 


limit  — ^ -" 


limit 


^1- 


w. 


12  FUNCTIONS  OF   A   COMPLEX   VABIABLE. 

or 

limit  ^>^'<*»-^«>  =  limit  !:v '<'''-••>. 


Hence 


limit  (0,-  0,)  =  limit  {6,-  0,) ; 


and  the  images  in  the  ze^-plane  of  the  two  paths  traced  by  J3 
form  at  w^  an  angle  equal  to  that  at  2„  in  the  ^-plane.  Accord- 
ingly, if  2  be  supposed  to  trace  any  configuration   whatever 

in  a  portion  of  the  -s'-plane  in  which  — —  is  determinate  and  not 

equal  to  zero,  every  angle  in  the  image  traced  by  w  will  be 
equal  to  the  corresponding  angle  in  the  -a-plane.  If,  for  exam- 
ple, such  a  portion  of  the  -sr-plane  be  divided  into  infinitesimal 
triangles,  the  corresponding  portion  of  the  «;-plane  will  be 
divided  in  the  same  manner,  and  the  corresponding  triangles 
will  be  mutually  equiangular.  Such  a  copy  upon  a  plane,  or 
upon  any  surface,  of  a  configuration  in  another  surface  is  called 
a  "  conformal  representation." 


The  modulus  of  the  derivative     — -    :=  limit    ——    is  the 


dw        ,.    . 
-r-    =  hmit 
dz 

Aw 

Az 

"  magnification."     Its  value,  which,  in  general,  changes  from 
point  to  point,  may  be  obtained  from  the  relations 


dw 
d^ 


■={S)'+©'=(0+©' 


'~'  ^x    -dy        -dy  dx 

The  theory  of  conformal  representation  has  interesting  ap- 
plications to  map  drawing.* 

*For  the  literature  of  the  subject,  see  Forsyth,  Theory  oi  Functions, 
p.  500,  and  Holzmtiller,  Einfiihring  in  die  Theorieder  isogonalen  Verwandschat- 
ten  und  der  conforraen  Abbildungen,  verbunden  mit  Anwendungen  auf  mathe- 
matische  Physik. 


EXAMPLES   OF   CONFORMAL   REPRESENTATION. 


13 


Art.  9.    Examples  of  Conformal  Representation. 

Example  I. — Let  w  =  s -{- c.  This  function  is  formed 
from  the  independent  variable  by  the  addition  of  a  constant. 
Putting  for  zi/,  z^  and  ^,  respectively,  u  +  iVy  x  +  iy^  and  a  +  ib, 
one  obtains 

u-=.x-\-ay     v=y-\-d. 

Any  configuration  in  the  ^-plane  appears,  therefore,  in  the 
w-plane  unaltered  in  magnitude,  and  is  situated  with  respect  to 
the  axes  as  if  it  had  been  moved  parallel  to  the  axis  of  reals 
through  the  distance  a  and  parallel  to  the  axis  of  imaginaries 
through  the  distance  d.  The  following  diagrams  represent  the 
transformation  of  a  network  of  squares  by  means  of  the  rela- 
tion w  =  2  -}-  c. 


y 

X 


Example   II. — Let   w  =z  cs.     Writing  w  =  pe^y  zr=ire^^, 
and  c  =  r^e^^y  the  following  equations  result: 

The  origin  transforms  into  the  origin,  all  distances  measured 
from  the  origin  are  multiplied  by  a  constant  quantity,  and 
all  straight  lines  passing  through  the  origin  are  turned  through 
a  constant  angle.     See  the  following  diagrams. 


14 


FUNCTIONS  OF  A   COMPLEX   VARIABLE. 


Example  III. — Let  w  =  ^*.     Writings  ^  x-\-iy,  the  func- 
tion becomes 

w  =  e*^^  =  ^*(cos^  4-  /  sin^). 

Every  straight  line  x  =  t^  parallel  to  the  axis  of  y  is  trans- 
formed into  a  circle  p  =,  ^*  described  about  the  origin  as  a 
center,  the  axis  of  J  becoming  the  unit  circle.  Points  to  the 
right  of  the  axis  oi y  fall  without  the  unit  circle,  while  points 
to  the  left  of  this  axis  fall  within.  Every  straight  line  y  =  t^ 
parallel  to  the  axis  of  x  becomes  a  straight  line  v/u  ■=  tan  /, 
passing  through  the  origin.  The  accompanying  diagrams* 
exhibit  in  a  simple  manner  the  periodicity  expressed  by  the 

equation 

exp  {2  -f  2n7ti)  =  exp  (-3^), 

where  n  is  any  positive  or  negative  integer. 

To  every  point  in  the  w-plane,  excluding  the  origin,  corre- 
spond an  infinite  number  of  points  in  the  .^-plane.  These 
points  are  all  situated  on  a  straight  line  parallel  to   the  axis  of 


*The  figures  of  this  and  the  following  example  are  taken  from  Holzmilller's 
treatise. 


EXAMPLES   OF    CONFORMAL   REPRESENTATION. 


15 


y,  and  divide  it  into  segments,  each. of  length  2;r.    If  z'  be  one 
of  these  points,  the  general  value  of  the  inverse  function  is 

log  w  z=z  z'  Ar  2ni7r, 

where  n  is  any  positive  or  negative  integer. 

If  any  straight  line  beginning  at  the  origin  be  drawn  in  the 
ij£/-plane,  there  will  correspond  in  the  ^-plane  an  infinite  number* 


27t- 


of  straight  lines  parallel  to  the  axis  of  x,  dividing  that  plane 
into  strips  of  equal  width.  To  any  curve  in  the  w-plane 
which  does  not  meet  the  line  just  drawn,  will  correspond  in 
the  -s'-plane  an  infinite  number  of  curves,  of  which  there  will  be 
one  in  each  strip. 

Example  IV. — Let  w  =  'cos  z.  Writing  w  =  u-{-w,  z  = 
Xr\-iy,  and  employing  as  equations  of  definition  cos  (/r)  = 
cosh  J,  sin  {iy)  =  /  sinh  y,  the  given  function  takes  the  form 


Hence 


u  -f-  iv  ==  cos  X  cosh  y  —  t  sin  x  sinh  y. 
u  =  cos  X  cosh  y,  V  ^=  —  sin  ;r  sinh^. 


16 


FUNCTIONS   OF   A   COMPLEX   VARIABLE. 


Any  straight   line,  x  =  /,,  parallel  to.  the  axis  of  ^,  is  trans- 
formed into  one  branch  of  a  hyperbola,  ^^ 


=  I, 


cos'  /,      sin''  /, 

having  its  foci  at  the  points  +  i  and  —  i.      Any  straight  line, 
^  =  /, ,  parallel  to  the  axis  of  x,  is  transformed  into  an  ellipse, 


+ 


=  I, 


cosh'  /,      sinh'  /, 

having  its  foci  at  the  same  points,  any  segment  of  the  straight 
line  equal  in  length  to  27t  corresponding  to  the  entire  curve 
taken  once.  By  means  of  these  confocal  conies,  the  w-plane 
is  divided  into  curvilinear  rectangles,  the  conformal  represen- 
tation breaking  down  only  at  the  foci,  where  the  condition 
dw 


that  ~  should   be  different   from   zero  is  not   fulfilled. 
az 

periodicity  of  the  function,  expressed  by  the  equation  . 
COS(<S'  +  271)  —  cos-s,  r 


The 


) 

y 

15 

16 

1 

3 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

U 

15 

16 

1. 

2 

6 

P 

A 

B 

C 

D 

E 

F 

Q 

H 

/ 

J 

K 

L 

M 

N 

0 

P 

A 

B 

X 

B 

A 

F 

0 

N 

M 

L 

K 

J 

I 

H 

G 

F 

E 

D 

C 

B 

A 

P 

0 

2 

1 

16 

15 

14 

13 

13 

11 

10 

9 

8 

7 

6 

5 

i 

3 

2 

1 

16 

15 

\ 

is  exhibited  graphically 
in  the  accompanying 
diagrams. 

It  is  interesting  to 
note  in  this  example, 
as  also  in  the  preceding 
one,  that  the  conformal 
representation  intro- 
duces well-known  sys- 
tems of  curvilinear 
coordinates,  the  cartesian   coordinates,  x,  f  o(  a  point  in  the 


EXAMPLES    OF    COifFORMAL   REPRESEN^TATION". 


17 


^-plane  serving  to  determine  its  image  in  the  ze/-plane  as  an 
intersection  of  orthogonal  curves. 

Example  V. — Let  w  =  z^.  Writing  w  =  «  +  «/,  ^  = 
X  +  iy^  the  relations 

«  u  =  x^  ^  s^y^t  'v  =  s^y  —  y 

follow  at  once.  If  one  of  the  variables  x,j/  be  eliminated  from 
these  two  equations  by  means  of  the  equation  /x  -\-  my  -\-n  =  Oy 
representing  a  straight  line  in  the  ^--plane,  equations  are  ob- 
tained representing  a  unicursal  cubic  in  the  w-plane. 

By  putting  w  =  p{cos  (p  -{-  i  sin  0),  z  =  r(cos  0  -{-  t  sin  6\ 
the  relations  p  =  r\  0  =  36^,  are  obtained.  Hence  the 
circle 

r'  —  2ar  cos  6  -{•  a*  =  c* 

gives  the  curve 

pi  —  2ap^  cos  —  -f  ^'  =  ^, 

which  enwraps  three  times  the  point  corresponding  to  the 
center.  The  accompanying  figure  represents  this  transfor- 
mation, the  straight  Hne  /<?^  giving  the  curve  /e^. 


div 


To  each  point  in  the  w-plane,  excluding  the  origin,  at  which 
=  O  and  the  conformal  representation  is  not  maintained, 


18 


FUNCTIONS  OF  A   COMPLEX  VARIABLE. 


there  correspond  three  points  in  the  -s^-plane,  having  for  their 
0  0  -f-  2;r  0  -f-  4?r 


arguments  -, 


respectively.    Any  straight  Hne 


drawn  from  the  origin  in  the  w-plane  will  have,  therefore,  three 
images  in  the  -s^-plane,  viz.,  three  straight  lines  diverging  from 
the  origin,  and  dividing  the  plane  into  three  equal  regions. 
Any  continuous  curve  in  the  w-plane  not  meeting  the  line  just 
drawn  will  be  represented  in  the  ^-plane  by  three  curves,  of 
which  one  will  be  situated  within  each  of  these  regions.  In  the 
figure  here  given  are  exhibited  the  three  conformal  represen- 
tations of  a  square  formed  in  the  7£;-plane  by  lines  u  =  t^,  u  = 
t^,v=t^^v  =  t^y  parallel  to  the  axes. 

If  the  relation  between  w  and  z  be  reversed,  and  z  be 
taken  as  a  function  of  w,  z  will  be  a  three-valued  function,  its 
values  giving  rise  to  three  branches  which  will  remain  distinct 
and  continuous  except  when  w  becomes  equal  to  zero. 


d      c 
a      h 


Prob.  8.  U  w  =  z  -\ ,  show  that  circles  in  the  ^-plane  having 

z 

a  common  center  at  the  origin  transform  into  confocal  ellipses. 


Prob.  9.  If  IV 


;.  show  that  the  axis  of  reals  in  the  s'-plane 


transforms  into  the  circle  \7v\  =  i,  and  the  upper  half  of  the  ^-plane 
into  the  interior  of  this  circle. 


COKFORMAL   REPRESENTATION^    OF   A   SPHERE. 


19 


Art.  10.  CoNFORMAL  Representation  of  a  Sphere. 

Let  OPO'  be  a  sphere  having  its  diameter  00'  equal  in 

length  to  unity.  Con- 
struct tangent  planes  at 
at  6>  and  O' .  Draw  in 
the  tangent  plane  at 
O  rectangular  axes  Ox 
and  Oy ;  and  in  the 
other  plane  draw  as 
axes  O'u,  parallel  to  Ox 
and  measured  in  the 
same  direction,  and  O'v 
parallel  to  Oy  but  meas- 
ured in  a  contrary  di- 
rection. Join  any  point 
z  in  the  plane  xOy  to 
C  by  a  straight  line,  and  let  O'z  meet  the  sphere  in  P.  Draw 
(9Pand  produce  it  to  meet  the  plane  uO'v  in  w. 

From  the  similar  triangles  O' Oz  and  OO'w 


Oz 
00' 


00' 
O'w 


that  is, 


or     Oz .  O'w  =  00' 


w\  =  rp=  I, 


To  an  observer  standing  on  the  sphere  at  O'  rotation  about 
OO'  from  O'lc  toward  O'v  is  positive,  while  to  an  observer 
standing  on  the  sphere  at  O  such  a  rotation  is  negative. 
Hence 

Z_xOz  =  —  Z^O'w,    or     6^  =  —  0. 

The  following  equation  results : 


wz 


pre 


t(<f>  +  0) 


The  W'  and  ^-planes  are  therefore  conformal  representa- 
tions of  one  another.  Any  configuration  in  one  plane  can  be 
formed  from  its  image  in  the  other  by  an  ijiyersjon  with  respect 


20  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

to  the  origin  as  a  center,  combined  with  a  reflection  in  the  axis  of 
reals.  Such  a  transformation  was  termed  by  Cayley  a  "  quasi- 
inversion."  By  it  points  at  a  great  distance  from  the  origin 
in  one  plane  are  brought  near  together  in  the  immediate  neigli- 
borhood  of  the  origin  in  the  other  plane. 

"Since  the  Hne  O' Pz  makes  the  same  angle  with  the  plane 
tangent  to  the  sphere  at  P  as  with  the  plane  xOy^  any  spherical 
angle  having  its  vertex  at  P  is  projected  into  an  equal  angle  at 
z.  The  sphere  is  thus  seen  to  be  related  conformally  to  the 
plane  xOy,  and  it  must  be  also  so  related  to  the  plane  uO'v, 

The  representation  of  the  sphere  upon  a  tangent  plane  in 
the  manner  described  above  is  termed  a  "stereographic  pro- 
jection." When  to  this  representation  is  applied  a  logarithmic 
transformation,  that  is,  one  inverse  to  the  transformation 
described  in  Example  III  of  the  preceding  article,  the  so- 
called  *  *  Mercator's  projection  ' '  is  obtained. 

Art.  11.    Conjugate  Functions. 

The  real  and  imaginary  parts  of  a  monogenic  function, 
ze;  =  «  4"  ^^>  have  been  shown  to  satisfy  the  partial  differential 
equations 

a«  _  a^       'd;v__  _'du 

dx  ~  dy  'dx  ~       dy' 

At  any  point,  therefore,  where  u  and  v  admit  second  partial 
derivatives,  one  obtains 

a^  ,  a^_         av  ,  d'v _ 
dx^'^dr      '      dx'~^dy      ' 

that  is,  the  functions  u  and  v  are  solutions  of  Laplace's  equa- 
tion for  two  dimensions.  Any  two  real  solutions  /  and  ^  of 
this  equation,  such  that  p-\-iq  is  a  monogenic  function  of 
X  +  iy,  are  called  "  conjugate  functions."  *  Thus  the  examples 
of  Art.  9  furnish  the  following  pairs  of  conjugate  functions: 

*  Maxwell,  Electricity  and  Magnetism,  1873,  vol.  i,  p.  227. 


applicatio:n'  TO  fluid  motion".  21 

x  -\-  a,  y  -\-  d  ]  r^r  cos  {B^  +  ^)»  ^i^  sin  (6^,  ■\-B)\  ^*  cos  j,  e^  s\x\y ; 
cos  X  cosh  J,  —  sin  x  sinhj  ;  x^  —  3^jk',  Z^y^  —  y^-  Tlie  second 
pair  is  expressed  in  polar  coordinates,  but  may  be  transformed 
to  cartesian  coordinates  by  means  of  the  relations 

r  =   Vx'  +/,     cos  e  =  -_^  sin  6 


Vx'+f  ,    Vx'+f 

If  one  of  two  conjugate  functions  be  given,  the  other  is 
thereby  determined  except  for  an  additive  constant.  Let  Uy 
for  example,  be  given.     Then 

,         'dv  y     .    dVy 
dv  =  — ax  4-  —ay 

'dx         dy 


— ax  A ay. 

dy     ^  dx  -"' 


and  therefore  the  value  of  v  is 


/( 


dy     ^  dx-"^ 


The  equations  u  =^  c^,  v  =^  c^,  obtained  by  assigning  con- 
stant values  to  two  conjugate  functions,  represent  in  the 
7^-plane  straight  lines  parallel  to  the  coordinate  axes.  It 
follows  that  the  curves  which  these  equations  define  in  the 
^-plane  intersect  at  right  angles.  Consequently,  by  varying 
the  quantities  c^  and  ^,,  two  orthogonal  systems  of  curves  are 
obtained  ;  and  c^  and  c^  may  be  taken  as  orthogonal  curvilinear 
coordinates  for  the  determination  of  position  in  the  ^-plane. 

Prob.  lo.  Show  that  if  /  and  ^  are  conjugate  functions  of  u  and 
V,  where  u  and  v  are  conjugate  functions  of  x  andj,  /  and  ^  will  be 
conjugate  functions  of  x  and  y. 

Prob.  II.  Show  that  if  u  and  v  are  conjugate  functions  of  x  and 
y,  X  and^  are  conjugate  functions  of  u  and  v. 

Art.  12.    Application  to  Fluid  Motion. 

Consider  an  incompressible  fluid,  in  which  it  is  assumed 
that  every  element  can  move  only  parallel  to  the  ^-plane,  and 
has  a  velocity  of  which  the  components  parallel  to  the  coordi- 


22  FUNCTIONS   OF  A   COMPLEX   VARIABLE. 

nate  axes  are  functions  of  x  and  ^  alone.  The  whole  motion 
of  the  fluid  is  known  as  soon  as  the  motion  in  the  ^-plane  is 
ascertained.  When  any  curve  in  the  ^-plane  is  given,  by  the 
"flux  across  the  curve""*  will  be  meant  the  volume  of  fluid 
which  in  unit  time  crosses  the  right  cyHndrical  surface  having 
the  curve  as  base  and  included  between  the  ^-plane  and  a  par- 
allel plane  at  a  unit  distance. 

The  flux  across^  any  two  curves  joining  the  points  js^  and  2 
IS  the  same,  provided  the  curves  enclose  a  region  covered  with 
the  moving  fluid.  For,  corresponding  to  the  enclosed  region, 
there  must  be  neither  a  gain  nor  a  loss  of  matter.  Let  s^  be 
fixed,  and  2  be  variable.  Let  ip  denote  the  flux  across  any  curve 
s^2,  reckoned  from  left  to  right  for  an  observer  stationed  at  2^ 
and  looking  along  the  curve  toward  2.  If  /,  m  be  the  direction 
cosines  of  the  normal  (drawn  to  the  right)  at  any  point  of  the 
curve,  and  /,  g  be  the  components  parallel  to  the  axes  of  the 
velocity  of  any  moving  element,  the  value  of  ip  will  be 


where  the  path  of  integration  is  the  curve  joining  z^  and  z. 
The  function  ^  is  a  one-valued  function  of  z  in  any  region 
within  which  every  two  curves  joining  z^  to  z  enclose  a  region 
covered  with  the  moving  fluid. 

If  z  moves  in  such  a  manner  that  the  value  of  rp  does  not 
vary,  it  will  trace  a  curve  such  that  no  fluid  crosses  it,  i.e.,  a 
"  stream-line."  The  curves  ^  =  const,  are  all  stream-lines,  and 
^  is  called  the  "  stream-function."  If  p  and  q  are  continuous, 
and  if  z  be  given  infinitesimal  increments  parallel  to  x  and  y 
respectively,  one  obtains 

'dx~      ^'      -dy     ^* 
If  now  the  motion  of  the  fluid  be  characterized,  as  is  the 

*  Lamb's  Hydrodynami  s  (1895).  p.  6q. 


APPLICATION   TO   FLUID   MOTION. 


23 


case  in  the  so-called  "  irrotational"  motion  *  by  the  existence 
of  a  velocity-potential  0,  so  that 

^      90  90 

the  following  equations  result : 

a0_a^    9^_  _a0 

Hence  0  +  /^  is  a  monogenic  function  of  ;tr  +  iy.  The  curves 
0  =  const.,  which  are  orthogonal  to  the  stream-lines,  are 
called  the  "  equipotential  curves." 

Consider,  as  an  example,  the  motion  corresponding  to  the 
functionf  w  =  z^.     The  equipotential  curves  are  given  by  the 

equations 

«  =  ;tr* — 3;ry = CO  n  St. , 

the  stream-lines  by 
the  equations 

V  =  'iyx'y  —y^^=  const. 

In  the  following  fig- 
ure the  stream-lines 
are  the  heavy  lines, 
while  the  equipo- 
tential curves  are 
dotted. 

The   fluid   moves 
i  in  toward  the  origin, 

which  is  called  a  '•  cross-point,"  from  three  directions,  and 
flows  out  again  in  three  other  directions.  At  the  cross-point 
the  fluid  is  at  a  standstill,  since  at  that  point  the  velocity,  for 
which  the  general  expression  is 


\/(g)'+(i)'. 


*  In  irrotational  motion  each  element  is  subject  to  translation  and  pure 
strain,  but  not  to  rotation. 

f  F.  Klein  :  Riemann's  Theory  of  Algebraic  Functions  ;  translated  by 
Frances  Hardcastle  (1893),  p.  3. 


24 


FUNCTIONS   OF   A   COMPLEX   VARIABLE. 


is  equal  to  zero.  The  stream-lines  in  the  figure  represent  the 
motion  of  the  fluid  in  each  of  six  different  angles,  as  if  the  fluid 
were  confined  between  walls  perpendicular  to  the  ^-plane. 

It  is  of  importance  to  note  that  if  the  function  considered 
be  multiplied  by  i,  the  equipotential  curves  and  stream-lines 
are  interchanged,  since  the  function  (p -\- tip  then  becomes 
—  ^  +  i(t>. 

An  example  of  particular  interest  is 


w 


—  //log 


+  ^ 


Let  z  —  a  =^  ^/'^',  2  -\-  a  =^  r^e'^^ ;  then 


«  =  —  //  log  -i,     v=  —  ^{d,  —  e,). 


The  curves  u  =  const.,  v  =  const,  form  two  orthogonal  sys- 
tems of  circles,  either  of  which  may  be  regarded  as  the  stream- 
lines, the  other  constituting  the  equipotential  curves. 


The  velocities  are  everywhere,  except  at  the  points  ±  ^. 
finite  and  determinate.  If  the  circles  r,/r,  =  const,  be  taken 
as  the  stream-lines,  each  of  the  points  ±  ^  is  a  "  vortex-point." 
If  the  circles  6*,  -  ^,  =  const,  be  taken  as  the  stream-lines,  one 


1^ 


SINGULAR   POINTS.  25 

of  the  points  ±  a  is  a  "  source,"  the  other  a  "sink."  In  the 
latter  case,  besides  the  hydrodynamical  interpretation,  a  very 
sinnple  electrical  illustration  is  afforded  by  attaching  the  poles 
of  a  battery  to  a  conducting  plate  of  indefinite  extent  at  two 
fixed  points  of  the  plate.  " 

As  another  example  may  be  taken  the  relation  w  =  cos  ^. 
As  has  been  shown,  the  curves  x  =  const,  form  a  system  of 
confocal  hyperbolas,  while  the  curves  ^  =  const,  form  an 
orthogonal  system  of  ellipses.  Either  system  may  be  regarded 
as  stream-lines.  In  one  case  the  motion  of  the  fluid  would  be 
such  as  would  occur  if  a  thin  wall  were  constructed  along  the 
axis  of  reals,  except  between  the  foci,  and  the  fluid  should  be 
impelled  through  the  aperture  thus  formed.  In  the  other  case 
the  fluid  would  circulate  around  a  barrier  placed  on  the  axis  of 
reals  and  included  between  the  foci. 

Besides  their  application  to  fluid-motion,  conjugate  func- 
tions have  important  applications  in  the  theory  of  electricity 
and  magnetism  *  and  in  elasticity .f 

Art.  13.    Singular  Points. 

Let  w  be  any  rational  function  of  js.  It  can  be  written  in 
the  form 

«'-0(^)' 

where /(2')  and  (p  {£)  are  entire  and  without  common  factors. 
This  function  is  finite  and  admits  an  infinite  number  of  suc- 
cessive derivatives  for  every  finite  value  of  ^,  except  the  roots 
of  the  equation  0  (-S")  =  o.  Let  a  be  such  a  root.  Then  the 
reciprocal  of  the  given  function  is  finite  and  admits  an  infinite 
number  of  successive  derivatives  at  the  point  a.     Such  a  point 

*  J.  J.  Thomson,  Recent  Researches  in  Electricity  and  Magnetism  (1893), 
p.  208. 

\  Love,  Theory  of  Elasticity  (1892),  vol.  i,  p.  331. 


26  FUNCTIONS   OF   A    COMPLEX   VARIABLE. 

is  called  a  "pole."  Any  rational  function  having  a  pole  at  a 
can  be  put  by  the  method  of  partial  fractions  in  the  form 

where  A^y  . .  .^  A,,  are  constants,  A^  being  different  from  zero, 
and  tl){z)  is  finite  at  the  point  a.  The  integer  k  is  said  to  be 
the  "order'*  of  the  pole,  and  the  function  is  said  to  have  for 
its  value  at  a  infinity  of  the  >^th  order.  In  accordance  with 
the  definition  of  a  derivative,  w  does  not  admit  a  derivative  at 
a.  From  the  character  of  the  derivative  in  the  immediate 
neighborhood  of  a,  however,  the  derivative  is  sometimes  said 
to  become  infinite  at  a. 

The  trigonometric  function  cot^  has  a  pole  of  the  first 
order  at  every  point  z  =  nnty  m  being  zero  or  any  integer  posi- 
tive or  negative. 

The  function  w  =  log  (2'  —  a)  has  for  every  finite  value  of 
2,  except  z  =  a,  a.n  infinite  number  of  values.  U  z  —  a  is  writ- 
ten in  the  form  Re*®, 

w  —  log  R  -\-  z{©  +  2m7r)f 

where  log  R  is  real,  and  m  is  zero  or  any  positive  or  negative 
integer.  If  3  describes  a  straight  line,  beginning  at  a,  S  will 
remain  fixed,  but  R  will  vary.  The  images  in  the  w-plane  will 
therefore  be  straight  lines  parallel  to  the  axis  of  reals,  dividing 
the  plane  into  horizontal  strips  of  width  27t.  If  now  the  ^--plane 
is  supposed  to  be  divided  along  the  straight  line  just  drawn, 
and  z  varies  along  any  continuous  path,  subject  only  to  the 
restriction  that  it  cannot  cross  this  line  of  division,  there  will 
be  a  continuous  curve  as  the  image  of  the  path  of  z  in  each 
strip  of  the  2£/-plane.  Each  of  these  images  is  said  to  corre- 
spond to  a  "branch"  of  the  function,  or,  expressed  otherwise, 
the  function  is  said  to  have  a  branch  situated  in  each  strip. 
The  line  of  division  in  the  ^-plane,  which  serves  to  separate 
the  branches  from  one  another  is  called  a  "  cut." 


3IKGULAR   POIN-TS.  27 

At  the  point  2  =  a  no  definite  value  is  attached  to  the 
function.  As  z  approaches  that  point  the  modulus  of  the  real 
part  of  the  function  increases  without  limit,  while  the  imagi- 
nary part  is  entirely  indeterminate. 

Let  2^  be  an  arbitrary  point,  distinct  from  a,  and  let 

log  R^  +  t&^  +  2m7rt 

be  any  one  of  the  corresponding  values  of  the  function.  Sup- 
pose that  2  starts  from  2^  and  describes  a  closed  path  around 
the  point  a,  the  values  of  the  function  being  taken  so  as  to 
give  a  continuous  variation.  Upon  returning  to  the  point  2^ 
the  value  of  the  function  will  be 

log  R,  +  /©„  +  2{m  +  i)7[t, 
or  log  R,  +  iQ^  -{-2(m—  i)7ti, 

according  as  the  curve  is  described  in  a  positive  or  negative 
direction.  By  repeating  the  curve  a  sufficient  number  of  times 
it  is  evidently  possible  to  pass  from  any  value  of  the  function 
at  z^  to  any  other.  When  a  point  is  such  that  a  ^-path  en- 
closing it  may  lead  in  this  manner  from  one  value  of  a  function 
to  another  value,  it  is  called  a  "  branch-point."  In  the  case 
of  the  function  here  considered,  the  point  z  =1  a  is  called 
a  "logarithmic  branch-point,"  or  a  point  of  "logarithmic 
discontinuity." 

The  function  w  =  log  ^^-)-{,  where  /{z)  and  (p{z)  are  entire, 

(p{z) 

has  a  point  of  logarithmic  discontinuity  at  every  point  where 
either /(^)  or  (p{z)  is  equal  to  zero.     For,  writing 

/{z)  =  A{z  -  ay^{z  -  a.y^  .  .  . 
0(^)  =  B{z  —  b,)^^{z  —  ^,)^» ... 

the  value  of  w  may  be  written 

A 
w  =  log  -B  +  ^">-  log  {z  —  a^)  -  ^q^  log  {z  —  hi), 

£>  rn.  n 


28  FUNCTIONS   OF  A   COMPLEX   VARIABLE. 

Take  now  the  function  w  =  e'*.  It  has  a  single  finite  value 
for  every  value  of  z  except  2  =  0.  If  ^  is  supposed  to  ap- 
proach zero,  the  limit  of  the  value  of  the  function  is  indeter- 
minate. 

For  let  /  +  iq  be  perfectly  arbitrary,  and  write 

If  now  a  +  lb  is  the  reciprocal  of  /  +  iq,  so  that 
—      /  A  _     —^ 

the  preceding  equation  may  be  written 

e^TTb  z=  c  -\-  id. 

But  whatever  the  value  of  the  integer  m,  q  -\-  imn  may  be 
substituted  for  q  without  altering  the  value  of  ^  +  id,  and  hence 
both  a  and  b  may  be  made  less  than  any  assignable  quantity. 

The  given  function  e^  therefore  takes  the  value  c  -\-  id  at  points 
a  +  ib  indefinitely  near  to  the  origin.  A  point  such  that,  when 
z  approaches  it,  a  fiinction  elsewhere  one-valued  may  be  made 
to  approach  an  arbitrary  value  is  called  an  •  *  essentiajl^  singu- 
larity." ^ 

Prob.  12.  Show  that  for  the  function  ^«^^  ^  =  dt  is  an  essei^ial 
singularity. 

Prob.  13.  The  function  e  ^*  considered  as  a  function  of  a  real 
variable  is  continuous  for  every  finite  value  of  z,  and  the  same  is 
true  of  each  of  its  successive  derivatives.  Show  that  when  it  is 
regarded  as  a  function  of  a  complex  variable,  -?  =  o  is  an  essential 
singularity. 

In  order  to  illustrate  still  another  class  of  special  points 
take  the  function 


w=s/{z  —  a^{z  —  «,)...  (^  —  ^«). 


SINGULAR   POINTS.  29 


ni 


This  function  has  at  every  finite  point,  except  <^j,  «,,...,  ^, 
two  distinct  values  differing  in  sign.  At  these  points,  however, 
it  takes  but  a  single  value,  zero.  From  each  of  the  points 
a^,  a^,  .  .  . ,  a„\et  a.  straight  line  of  indefinite  extent  be  drawn  in 
such  a  manner  that  no  one  of  them  intersects  any  other,  and 
suppose  the  ^s-plane  to  be  divided,  or  cut,  along  each  of  these 
lines.  Along  any  continuous  path  in  the  ^-plane  thus  divided 
the  values  of  the  function  form  two  distinct  branches. 

For,  writing 

js  —  a^  =  r/'*i,    z  —  a^z=z  r,^''*9,    . .  , ,    ^  —  a«  =  r,^**", 


the  function  takes  the  form 


A+h+  ...  +t 


w  =  Vr^r^  .  .  r„    e* 

,  No  closed  path  in  the  divided  plane  will  enclose  any  of  the 
points  a^,  a^,  .  .  .  ,  a„,  and  the  quantities  6^,  d,,  .  .  .  ,  0„,  after 
continuous  variation  along  such  a  path,  must  resume  at  the 
initial  point  their  original  values.  No  such  path,  therefore,  can 
lead  from   one  value  of  the  function  at  any  point  to  a  new 

value  of  the  function  at  the 
same  point.  If,  however,  the 
cuts  are  disregarded  and  s 
traces  in  a  positive  direction, 
a  closed  curve  including  an  odd 
number  of  the  points  a^,  ^,, 
.  .  .  y  a„,  and  not  intersecting 
itself,  then   an   odd  number  of 


the  quantities  O^y  6^,,  .  .  .  ,  ^„  are  each  increased  by  27r;  and 
the  value  of  the  function  is  altered  by  a  factor  ^(2*+i)t»^  and 
so  changed  in  sign.  In  the  same  way  any  closed  path  de- 
scribed about  one  of  these  points,  and  enwrapping  it  an  odd 
number  of  times,  leads  from  one  value  of  the  function  to 
the  other.  On  the  other  hand,  a  simple  closed  path  enclosing 
an  even  number  of  these  points,  or-  a  closed  path  which  en- 
closes but  one  of  the  points,  enwrapping  it  an  even  number  of 
times,  leads  back  to  the  initial  value  of  the  function.     It  fol- 


80  FUNCTIONS  OF  A   COMPLEX   VARIABLE. 

lows  that  each  of  the  points  ^, ,  ^, ,  .  .  .  ,  ^^  is  a  branch-point. 
Any  point  in  the  ^-plane,  closed  paths  about  which  lead  from 
one  to  another  of  k  set  of  different  values  of  a  function,  the 
number  of  values  in  the  set  being  finite,  is  called  an  "  algebraic 
branch-point." 

As  a  further  illustration,  consider  the  function 
w  =  2^-{-(2  —  a)\ 
which  is  a  root  of  the  equation  of  the  sixth  degree, 

w*  —S^w*  —  2(2  —  ayw"  +  S-s'w'  —  62(2  —  a)w  -\-  (2— a)^— 2*^0. 

The  function  has  at  every  point,  except  2  =  0  and  2  ^a^ 
six  distinct  values.  Six  branches  are  thereby  formed  which 
can  be  completely  separated  from  one  another  by  making  cuts 
from  the  points  2  =  0  and  2  =  a  to  infinity.  Putting  a>  for  the 
cube  root  of  unity,  these  six  branches  can  be  written 

1/2      ,  ,  vl/3  1/2      ,  ,  .1/3 

w^  =  2     -{-      {2  —  a)' ,     w^=  —  2     -^      {2  —  a)^ , 
w^  —  z^^  -{-  gd{2  —  d)'\     w^-=^  —  2^  -\-  gd{2  —  df^^, 

^  W^  =  2'^^  +  Q0^{2  —  dj^^y       Zf ,  =  —  2^^  +  ^\^  ~  ^)*^'» 

The  branches  w,  and  w^,  w^  and  w^,  w^  and  w^  are  interchanged 
by  a  small  closed  circuit  described  about  s  =  o,  while  a  small 
circuit  described  about  2-=  a  permutes  cyclically  the  branches 
w^,  w^y  w^y  and  also  the  branches  w^,  w^,  w^. 

All  of  the  special  points  examined  above,  poles,  points  of 
/  logarithmic  discontinuity,  essential  singularities,  and  branch- 
points, are  called  singular  points.  In  fact,  a  function,  or  a 
branch  of  a  function,  is  said  to  have  a  *  *  singular  point '  *  at  each 
point  where  it  fails  to  have  a  continuous  derivative,*  or  about 
which  as  a  center  it  is  impossible  to  describe  a  circle  of  deter- 
minate radius  within  which  the  function,  or  branch,  is  one- 
Valued. 

Any  point  not  a  singular  point  is  called  an  "  ordinary  point." 

*  Continuity  and,  therefore,  finiteness  of  the  function  are  implied  in  the 
existence  of  a  derivative. 


POINT   AT   INFINITY.  31 

An  ordinary  point  at  which  a  function  reduces  to  zero  is  called 
a  **zero"  of  the  function. 

If  in  a  certain  region  of  the  ^-plane  a  function  is  uniform 
and  has  no  singular  points,  the  function  is  said  to  be  ''synec- 
tic  "  or  *'holomorphic  "  in  that  region.  If  in  a  certain  region 
the  only  singular  points  of  a  uniform  function  are  poles,  the 
function  is  said  to  be  '*  meromorphic  "  in  that  region.  Under 
similar  conditions,  a  branch  of  a  function  is  also  described  as 
holomorphic  or  meromorphic. 

Prob.  14.  When  w  and  z  are  connected  by  the  relation  w  —  g  ^ 
(z  —  /lY  show  that  if  z  describes  a  circle  about  >^  as  a  center,  w 
describes  a  circle  about  g  as  a  center,  an  angle  in  the  ^-plane  hav- 
ing its  vertex  at  /i  is  transformed  into  an  angle  in  the  w-plane  t 
times  as  great  and  having  its  vertex  at  g^  and  that  2  =  ^  is  a  branch- 
point of  w  except  when  /  is  an  integer. 

Art.  14.    Point  at  Infinity. 

In  determining  the  limiting  value  of  a  function  when  the 
modulus  of  the  independent  variable  z  is  increased  indefinitely, 
it  is  usual  to  introduce  a  new  independent  variable  z'  by  the 
relation  z  —  \/z' ,  and  consider  the  function  at  the  point  z'  —  o. 
This  is  equivalent  to  passing  from  the  ^-plane  to  another  plane, 
the^^-plane,  related  to  the  former  by  the  geometrical  construc- 
tion described  in  Art.  10.  It  is  often  very  convenient,  however, 
to  go  further  and  to  suljgtitute  for  the  ^-plane  the  surface  of  the 
sphere  of  unit  diameter  touching  the  ^r-plane  at  the  origin.  No 
difficulty  is  thus  introduced  since,  as  explained  in  the  article 
just  cited,  any  configuration  in  the  -a-plane  obtains  a  conformal 
representation  upon  the  sphere;  and  the  advantage  is  gained 
that  the  entire  surface  upon  which  the  variation  of  the  inde- 
pendent variable  is  studied  is  of  finite  extent.  The  point  of 
the  sphere  diametrically  opposite  to  its  point  of  contact  with 
the  ^■-plane  coincides  with  the  point  written  above  as  z'  =  o. 
It  is  called  the  point  at  infinity,  5'  =  00 ,  since  a  point  on  the 
sphere  approaches  it  at  the  same  time  that  its  image  in  the 
.s:-plane  recedes  indefinitely  from  the  origin. 


32  FUNCTIONS   OF  A   COMPLEX  VARIABLE. 

The  point  at  infinity  may  be  either  an  ordinary  or  a  singular 

point.     For   the   function  ^,  for   example,  it  is  an    ordinary 

I 
point,  since  f»  =  e*' .     For  a  rational  entire  function  of  the  «th 

degree  it  is  a  pole  of  order  «.  Consider  it  for  the  function 
^{z  —  a^){z  —  a^  . .  .{z  —  a^),  discussed  in  the  preceding  article. 
Let  a  circle  of  great  radius  be  described  in  the  ^-plane  inclosing 
all  the  branch-points  ^, ,  tf,,  .  .  .  ,  ^„.  Its  con  formal  representa 
tion  on  the  sphere  will  be  a  small  closed  curve  surrounding  the 
point  <s'=  00.  This  point  must,  therefore,  be  regarded  as  a 
branch-point  or  not,  according  as  the  function  changes  value  or 
not  when  the  curve  surrounding  it  is  described,  that  is  accord- 
ing as  «,  the  number  of  finite  branch-points,  is  odd  or  even. 
When  the  point  at  infinity  is  taken  into  account,  then,  the 
total  number  of  branch-points  of  this  function  is  always  even. 
The  character  of  the  point  ^  =  00  for  this  function  can  be  de- 
termined directly,  by  changing  z  into  i/z'  and  considering  the 
point  2'  =  o. 

Prob.  15.  Show  that  0  =  00  is  an  ordinary  point  for  ~rr4j  where 

<p{z)  and  fp{z)  are  rational  and  entire  if  the  degree  of  (p{z)  does 
not  exceed  that  of  tp(z)» 

Art.  15.    Integral  of  a  Function. 

Let  w=/(z)  be  a  continuous  function  of  ^  in  a  given 
region,  and  suppose  z  to  describe  a  continuous  path  L  from 
the  point  z^  to  the  point  Z.  Let  a  series  of  points  z,,  z^,  . . .  ,-^„ 
be  taken  on  Z,  and  let  Z^,  /j, . . . ,  t„  be  points  arbitrarily  chosen 
on  the  arcs  z^z^,  z^z^, . . . ,  z^Z  respectively.     Form  the  sum 

s={z,-  z:)At:)  +  (^,  -  ^,  wo  + . . .  +  (^  -  ^j/(/«). 

If  now  the  number  of  points  z^y  . . . ,  ^„  be  increased  indefi- 
nitely in  such  a  manner  that  the  length*  of  each  of  the  arcs 

*  It  is  assumed  in  regard  to  every  path  of  integration  that  the  idea  of  length 
maybe  associated  with  the  portion  of  it  included  between  any  two  of  its  points, 
or,  what  is  the  same  thing,  that  the  path  is  rectifiable.  This  condition  is  evi- 
dently satisfied  if  the  current  coordinates  x  and;)/  can  be  expressed  in  terms  of 


IKTEGKAL  OF  A   FUNCTIOK. 


33 


^^3^fS^JS^f  ,,,yZnZ  approaches  zero  as  a  limit,  the  sum  5  ap- 
proaches a  finite  Hmit  which  is  inde- 
pendent of  the  choice  of  the  points  z^^ 
^„  ...,^„and  t,,  t^,...,  /«. 

For  take  any  other  sum 

5'  =  (v-^.)y(0+ 

formed  in  a  similar  manner.  Suppose 
"  for  the  sake  of  greater  definiteness 
and  z/f . . .  follow  one  another  on  the 


that  the  points  z^ , 
line  L  in  the  order 

Z^f  Z^  ,  Z^  f  Z^f  Z^,  Z,  f  ,  ,  ,  f 

and  form  a  third  sum 

in  which  b  )th  series  of  points  occur.  It  may  be  shown  that  as 
the  number  of  points  in  each  of  the  series  -sr,,  . . .  and  z/,  ...  is 
increased,  the  differences  S'^  —  S  and  S^'  —  S'  both  approach 
zero,  from  which  it  follows  that  the  difference  5  —  5' has  a 
limit  equal  to  zero.  For  example,  the  difference  5''  —  5  has 
the  value 

(^.  -  ^.)[/(r.)  -/('.)]  +  W  -  ^,)[/(r.)  -/(/,)] 
If  M  denotes  the  upper  limit  or  bound  of  the  quantities 

|y(n)-AOI.       \Ar,)-M)U       \A-',)-M)\ 

the  modulus  of  5"  —  5  will  be  less  than 

dx  dy  .  „ 

any  parameter  /  so  that  —   and  j-  are   continuous.      For  then  the  integral 

/    ^ dx''  +  dy*  is  finite.     See,  in  this  connection,  Jordan,  Cours  d'Analyse,  2d 
Edition,  Vol.  I.,  p.  lOO. 


34  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

But  1^,  —  -s-jI  is  equal  to  the  chord  of  the  arc  s^o-S', ,  and  must 
therefore  be  less  than  or  equal  to  this  arc,  and  a  similar  result 
holds  for  each  of  the  quantities  | z^  —z^  \A^\  ~^x  I » •  •  •     Hence 

\S"  -S\<Ml. 

where  /  denotes  the  length  of  the  path  of  integration.  When 
the  number  of  points  of  division  on  the  line  L  is  increased,  the 
differences 

/^^)  -  /('.).      A^^  -  At.)<      /(n)  - /(/,), . . . 

approach  zero,  since  f{z)  is  continuous.*  M  accordingly 
decreases  indefinitely  and  the  difference  S"  —  S  approaches 
zero. 

The  limit,  the  existence  of  which  has  just  been  demon- 
strated, is  called  the  integral  of  f\z)  along  the  path  L.     It  is 

written    I  f{z)dz.     The  definition  here  given  is  similar  to  that 

given  for  the  integral  of  a  function  of  a  real  variable.  It  is 
unnecessary  to  specify  the  path  of  integration  when  the  inde- 
pendent variable  is  restricted  to  real  values,  since  in  that  case 
it  must  be  the  portion  of  the  axis  of  reals  included  between 
the  limits  of  integration. 

The  following  well-known  principles,  applicable  to  the  case 
of  a  real  independent  variable,  may  be  readily  extended  to  the 
general  case : 

1.  The  modulus  of  the  integral  cannot  exceed  the  length  of 
the  path  of  integration  multiplied  by  the  upper  bound  of  the 
modulus  of  the  function  along  that  path. 

2.  The  independent  variable  may  be  altered  by  any  equa- 
tion of  transformation,  but  L' ,  the  path  of  integration  in  the 
transformed  integral,  must  be  such  that  it  is  described  by  the 
new  variable  while  z  describes  L. 

3.  If  F{z)  is  any  one-valued  function  having  everywhere 
a  continuous  function  f{z)  for  its  derivative,  the  equation 

must  be  true. 

*  For  a  complete  discussion  it  should  be  shown  that  the  continuity  oif{z)  is 
necessarily  '•uniform."     See  Jordan,  Cours  d'Analyse,  2d  Edition,  vol.  i,  p.  183. 


INTEGRAL   OF   A   FUNCTION.  35 

To  prove  the  third  principle,  write  F{Z)  —  F{z^  in  the 
form 

Since  the  derivative  of  F{z)  \sf{z)y 

F{2,„+;)  -  F{2„,)  =   [/(^,„)  +  ^J(^«.+,  -  ^m), 

where  //,„  has  zero  for  its  limit  *  when  z^j^^  is  made  to  approach 
z^.     Hence 

F{Z)  -  F{z,)  =  limit  2f{z,,){^^^.  -  ^n)  +  limit  :2'//,«(^„+,  -  z^) ; 

or,  since  the  second  term  of  the  right-hand  member  is  equal 
to  zero, 

F{Z)-F{.,)  =  J[Az)dz. 

If  no  function  F{z)   fulfilling  the  preceding  conditions  is 
known,    the   value    of    the    integral    requires    further   investi- 

Consider  as  an  example  the  integral   j  — ^  taken  from  the 

point  ^  =  —  I  to  the  point  z  =  i,  the  path  of  integration  being 
the  upper  half  of  the  circumference  of  a  unit  circle  described 
about  the  origin  as  a  center.  Writing  z  =  exp  {t6),  z  will 
describe  the  required  path  while  6  varies  from  tt  to  o. 

The  equations  -,  =  e'""'^,        dz  =  ie^^dd, 
z 

dz 

—  =  ie-^^dB  =  i  cos  d  dd -\-  sin  6  dS  ^  id  (sin  B)  —  d  (cos  B\ 

Z" 

follow  at  once.     Hence  for  the  path  specified 

J  —^-=1  Cd  (sin  B)  —    Cd  (cos  <9)  =  —  2. 

~^  It  n 

The  application   of  the  direct  and  more  famihar  method 
i?'ves  the  same  result: 

+  ^  J  r-  -I  r-  -1 

dz       III  III 

2. 


J  z'     L    zj,=-i    L    zj^^^i 


1 

*The    "uniformity"   of    continuity   is  involved   here.       See  Jordan,    Cours 

d' Analyse,  2d  Edition,  vol  i,  p.  184. 


36  FUNCTIONS  OF  A   COMPLEX   VARIABLE. 

For  a  path  along  the  axis  of  reals  between  the  limits  of 
integration  this  result  is  unintelligible.     The  discontinuity  of 

the  differential,  —i,  at  the  point  2  =  0,  prevents  the  considera- 

JS 

tion  of  such  a  path ;  and  that  the  result  should  be  negative 
when  the  differential  is  at  every  point  of  the  path  positive 
has  no  significance.  The  introduction  of  the  complex  variable 
furnishes  a  perfectly  satisfactory  explanation  of  the  result. 

Prob.  16,  Show  that  the  integral  of  —  along  any  senii-circum- 

z 

ference  described  about  the  origin  as  a  center  is  equal  to  tti. 


Art.  16.    Reduction  of  Complex  Integrals  to  Real. 
The  integral 

may  be  written  in  the  form 

or,  separating  the  real  and  imaginary  terms, 

/  {udx  —  vdjy)  -\-  i  I  {^dx  +  udy). 

Hence  the  calculation  of  the  integral  may  be  reduced  to 
the  calculation  of  two  real  curvilinear  integrals. 

The  equations 

9«  _  ?)V     'du  _  _'dv 
'dx  ~  d/     9j  dx 

which  express  the  condition  that  2/  +  ^^  should  be  monogenic, 
express  also  that 

udx  —  vdy,     vdx  -\-  udy 

are  the  exact  differentials  of  two  real  functions  of  the  variables 
X,  y.     Consider  the  case  where  these  functions  are  one-valued. 


cauchy's  theorem.  37 

Denoting  them  by  P(x,  y)  and  Q{x^  y)  respectively,  the  inte- 
gral may  be  written 

\_P(,X,  Y)  -  I\x„y:i\  +  i\_Q{X,  Y)  -  (2(^.,  jy.)], 

(•^o»Jo)  3^rid  (X,  F)  being  the  initial  and  terminal  points  re- 
spectively of  the  path  of  integration. 

Art.  17.    Cauchy's  Theorem. 

Cauchy's  Theorem  furnishes  the  necessary  and  sufficient 
conditions  that  a  uniform  function  /(z),  having  continuous 
partial  derivatives  with  respect  to  x  and  j',  should  yield  within 
a  region  bounded  by  a  continuous  closed  curve  a  one-valued 
integral,  that  is,  an  integral  the  value  of  which,  when  the  lower 
limit  is  fixed,  depends  simply  on  the  upper  limit,  and  not  on  the 
path  of  integration.  It  will  be  more  convenient,  before  consider- 
ing Cauchy's  Theorem,  to  demonstrate  the  following  lemma: 

Lemma. — Let  ^  be  a  portion  of  the  ^-plane,  having  a  bound- 
ary 5  which  consists  of  a  closed  curve  not  intersecting  itself, 
or  of  several  closed  curves  not  intersecting  themselves  or  one 
another.  If  at  every  point  of  the  region  A,  including  its 
boundary  5,  a  function  W  oi  the  real  variables  x  and  j  is  one- 
valued  and  continuous  and  has  continuous  partial  derivatives 

— — ,  — — ,  the  relations 

fm.  =  -Jl^,.,y  (a) 

•exist,  the  integrals  in  the  first  members  being  taken  along  the 
boundary  in  the  positive  direction,  and  those  in  the  second 
members  being  taken  over  the  enclosed  area. 

Denote  by  A  the  inclination  to  the  axis  of  x  of  the  exterior 
normal  at  any  point  of  the  boundary,*  that  is,  the  normal  drawn 

*  It  is  assumed  that  the  boundary  has  a  determinate  tangent  at  every  point. 
If  the  boundary  of  a  given  region  is  not  of  this  sort,  the  theorem  holds  for  any 
interior  curve  ot  which  this  assumption  is  true. 


36 


FUNCTIONS  OF  A   COMPLEX   VARIABLE. 


to  the  right  as  the  boundary  is  described  in  a  positive  direction. 
If  any  straight   line   parallel   to  the  axis  of  x  be  traced   in 
the  direction  of  increasing  values  of  x,  at  each  point  where 
it   passes  into   the   area  A^ 
cos  A  is  negative,  and  there- 
fore in  the  first  member  of 
{\)  dy=.  cos  \  (is  is  negative. 
At   each    point  where    this 
straight  line  passes   out  ol 
the  area  A,  cos  A,  and  there- 
fore dy^  in  the  first  member 
of  equation  (i),  is  positive. 
Hence  in  the  first  member 
of   equation  (i)   the   differ- 
entials   Wdy  corresponding 
to  a  given  value  of  y^  and  taken  in  the  order  of  increasing 
values  of  ;r,  have  signs  which,  compared  with  the  signs  of  the 
corresponding  values   of   W,  first    differ,  then  agree,   and   so 
on  alternately.     In  order  now  to  compare  the  integral  in  the 
first  member  of  equation  (i)  with  the  integral  in  the  second 
member,  it   is    necessary  to   take    dy  as    essentially   positive. 
The  sum  of  the  differentials  in  the  first   member,  correspond- 
ing to  a  fixed  value  of  jj/,  must  therefore   be  written  in  the 
form 

where  W^,  PF, ,  .  .  .  are  the  corresponding  values  of  W taken  in 
the  order  of  increasing  values  of  x.  But  performing  now  in 
the  second  member  of  equation  (i)  an  integration  with  respect 
to  X,  the  same  result  is  obtained,  so  that  the  two  members  of 
equation  (i)  become  identical,  and  the  equation  is  verified. 

To  obtain  equation  (2)  the  same  method  is  used.  It  is 
necessary  in  this  case  to  observe  that  if  a  line  parallel  to  the 
axis  of  jK  is  traced  in  the  direction  of  increasing  values  of  j^,  at 
each  point  where  it  enters  A,  dx  in  the  integral  of  the  first 


APPLICATION   OF   CAUCHY'S  THEOEEM.  39 

member  must  be  taken  as  positive;  and  at  each  point  where 
this  line  passes  out  o(  A,  dx  in  that  integral  must  be  taken  as 
negative. 

By  means  of  the  preceding  lemma,  Cauchy's  Theorem  is 
easily  proved.     This  theorem  may  be  stated  as  follows ; 

Theorem. — If,  on  the  boundary  of  and  within  a  given  region 
Aj  a  one-valued  function  w  =  f{z)  is  monogenic,  and  its  deriv- 
ative f{z)  is  continuous,*  the  integral  ff{z)dz  taken  along 
the  boundary  5  is  equal  to  zero. 

For  writing  the  integral  in  the  form 

J^wdz  =  J^{udx  —  vdy)  +  i  J{udy  +  vdx\ 
the  preceding  lemma  gives 

but  since  at  every  point  of  A 

the  given  integral  reduces  to  zero. 

Art.  18.    Application  of  Cauchy's  Theorem. 

From  Cauchy's  Theorem  it  follows  that,  if  two  different 
paths  Z,  and  Z,  lead  from  the  point  z^  to  the  point  Z,  and  if 
along  these  paths  and  in  the  region  inclosed  between  them  a 
given  function  f{z)  has  no  critical  points,  the  integrals  of  the 
function  taken  along  these  two  paths  are  equal.  For  two  such 
paths  taken  together,  one  described  directly,  the  other  re- 
versed, constitute  a  closed  curve,  and  the  integral  taken  along 

*  Otherwise  expressed,  the  one-valued  function /"(sr)  has  no  singular  points  on 
the  boundary  of  or  within  A,  (xf.z)  is  holomorphic  in  A.  It  has  been  shown  by 
Goursat  that  this  theorem  can  be  proved  without  assuming  the  continuity  of  the 
derivative.  ,  See  Transactions  Amer.  Math,  Soc,  vol.  I.  p.  14. 


40  FUNCTIONS   OF   A    COMPLEX   VARIABLE. 

it  is  equal  to  zero.  But,  since  reversing  the  direction  of  the 
path  of  integration  is  equivalent  to  changing  the  sign  of  the 
integral,  the  equation 

J^A^yz  -  f^Mdz  =  o 

is  obtained. 

The  result  just  established  may  be  stated  in  the  following 
theorem : 

Theorem  I. — If  a  function  is  holomorphic  in  any  simply 
connected  region  bounded  by  a  continuous  closed  curve,  the 
integral  of  the  function,  from  a  fixed  lower  limit  in  that  region 
to  any  point  contained  therein,  is  independent  of  the  path  of 
integration,  and  is  a  one-valued  function  of  its  upper  limit. 

A  region  whose  boundary  is  composed  of  disconnected 
curves  is  not  necessarily  characterized  by  the  property  stated 
in  the  theorem.     Take,  for  example,  the  function 


and  suppose  that  o  <  |^,  |  <  |^,  |  <  .  .  .  <  |  ^„  ].  With  the  ori- 
gin as  a  center,  construct  a  system  of  concentric  circles  (7,, 
^2,  •  .  •,  Cn,  C^  passing  through  ^,,  C^  through  ^„  and  so  on. 
Denote  by  5o  the  region  inclosed  within  the  first  circle  6',,  by 
S^  that  inclosed  between  C^  and  C,,  and  so  on,  the  portion 
of  the  plane  exterior  to  the  last  circle  C^  being  denoted  by  5«. 
At  an  initial  point  z^  interior  to  one  of  these  regions,  assign  to 
w  one  of  the  two  values  possible,  and  consider  the  branch  of 
w  resulting  from  a  continuous  variation.  Then  however  z  may 
vary  within  any  such  region,  this  branch  of  w  will  be  a  mono- 
genic function,  and  its  derivative  will  be  continuous.  Having 
regard  to  the  branch-points  ^,,  ^,,  .  .  .,  «„,  it  is. evident  that  in 
the  regions  5o,  5,  .  .  .  it  will  be  one-valued,  and  in  the  regions 
5j,  5,,  .  .  .  ,  it  will  be  two- valued.  Thus  in  the  regions  5,,  5^, 
. .  . ,  the  branch  fulfils  the  required  conditions,  but  the  boundary 
does  not.  The  theorem  is  applicable  only  to  S^.  It  may  be 
observed  that  in  every  other  region  two  paths  may  be  drawn 
joining  the  same  two  points  such  that  the  branch  is  not  one- 
valued  in  the  enclosed  portion  of  the  ^-plane. 


APPLICATION"   OF    CAUCHY^S    THEOREM.  41 

Theorem  II. — If/(-s')  is  holomorphic  in  any  simply  connected 
region  ^   bounded  by  a  continuous  closed  curve,  the  integral 

i  f{z)dz,  taken  from  a  fixed  lower  limit  z^  in  that  region  to  any 

point  Z  contained  therein,  is  a  holomorphic    function  of   its 
upper  limit. 

Let  L  be  any  path  from  z^  to  Z.  When  the  upper  limit  is 
at  the  point  Z  -\- dZ,  L  followed  by  a  straight  line  from  Z  to 
Z -\-  dZ  c^n  be  taken  as  the  path  of  integration.     Hence 

*Z+dZ  PZ  _     .    .  pZ+dZ 

*Z+dZ  nz+dz 


pz+dz  nz  pz-vdz 

pZ+dZ  pZ+dZ 


The  first  term  is  equal  to  f(Z)dZ.  The  modulus  of  second 
term  is  equal  to  or  less  than  M\dZ\j  where  M  is  the  upper 
bound  of  |/(z)  —  f{Z)\  along  the  line  joining  Z  to  Z-\-dZ, 
But  since  f{z)  is  continuous,  the  Hmit  of  J/ when  Z -\- dZ 
approaches  Z  is  zero.     Hence 

£'^'Az)dB  -  £j{z)dz  =  lAZ)  +  ^¥Z- 

where  rf  approaches  zero  with  dZ.     The  integral  therefore  has 
y(Z)fora  derivative,  and  is  holomorphic  in  S. 

In  the  case  of  a  region  bounded  by  several  disconnected 
closed  curves,  of  which  one  is  exterior  to  all  the  others, 
Cauchy's  Theorem  may  be  stated  in  the  following  form  : 

Theorem  III. — Let  a  function  f\z)  be  holomorphic  in  a 
region  A  bounded  by  a  closed  curve  ^  and  one  or  more  closed 
curves  C^,  C^,  .  .  .  interior  to  C.  The  integral  of  f{z)  taken 
along  C  will  be  equal  to  the  sum  of  its 
integrals  taken  in  the  same  direction 
along  the  curves  C^,  C^,  .  .  . 

For  the   integral  of  f{z)  taken  in  a 
positive  direction  completely  around  the 
boundary  of  A    is   equal    to  zero.      But 
tne  curves  (.7,,  C^,  .  .  .  are  then  described  in  the  direction  oppo- 


42  FUNCTIONS   OF  A    COMPLEX   VARIABLE. 

site  to  that  in  which  C  is  described.     Hence  if  all  the  curves 
are  described  in  the  same  direction,  the  result  may  be  written 

If  there  is  but  one  interior  curve,  so  that  the  region  A  is 
included  between  two  curves  C  and  C^,  the  integral  taken  along 
every  closed  curve  containing  C,  but  interior  to  C  has  the 
same  value,  viz.,  the  common  value  corresponding  to  the  paths 
C  and  6*,. 

Art.  19.    Theorems  on  Curvilinear  Integrals. 

Theorem  I. — li  fiz)  be  continuous  in  a  given  region  except 

at  the  point  a,  the  integral  j  f{z)dzy  taken  around  a  small  circle 

t,  having  its  center  at  a,  will  approach  zero  as  a  limit  simulta- 
neously with  the  radius  r  of  the  circle  c,  provided  only 

lim  {z  —  d)/{z)  =  o     when     z  =  a. 

For  let  the  upper  bound  of  the  modulus  of  {z  —  a)f(z)  on 
the  circle  c  be  denoted  by  M.     Then  at  every  point  of  c^ 

.  ^  X  -      M     _M 
mod  i\z)  -  '. r  -  — , 

and  consequently 

mod J^f{z)dz  %—J  ds  %.27tM. 

closed  curve  C  containing  the  point  a,  is  equal  to  zero,  except 
when  «  =  I.     When  «  =  i,  this  integral  is  equal  to  ini. 

For  the  value  of  the  integral  will  be  the  same  if  any 
circle  described  about  <?  as  a  center  be  taken  as  the  path  of 
integration.  Let  then  z  —  a  =:  r^'^  where  r  is  a  constant  and 
0  varies  from  o  to  2n.     The  integral  becomes 

-•  /»2»r     —  {n-\)ie 


THEOREMS   Oi^   CURVILINEAR   IKTEGRALS.  43 

which  reduces  to  zero  excepi  when  «  =  I.     If  «  =  i,  its  value 
is  2niy  whence 


/ 


27tl, 


Theorem  III. — If  f{z)  is  a  function  holomorphic  in  a  given 
region  5,  C  a  closed  curve  the  interior  of  which  is  wholly 
within  Sf  and  a  a  point  situated  within  (7,  then 


f  I^dz  =  2niAcf)' 


For  describing  about  <3;  as  a  center  a  small  circle  c  of  radius 
r,  the  equation 

*J^ z  —  a  ^'  2— a 

is  obtained.     But  at  every  point  of  c, 

where,  by  choosing  r  sufficiently  small,  the  modulus  of  tf  may 
be  made  less  than  any  fixed  positive  quantity.     Hence 

^cz—a  ^c2  —  a  *^c  z  —  a 

,"jut  by  the  preceding  theorems  the  first  term  of  the  right-hand 
member  is  equal  to  2nif{a),  and  the  second  term  is  equal  to 
zero. 

If  the  equation  of  the  theorem  just  established  be  differ- 
entiated with  respect  to  a,  the  following  important  formulas, 
expressing  the  successive  derivatives  of  a  holomorphic  function 
at  a  given  point,  are  obtained: 


44  FUNCTIONS  OF   A   COMPLEX   VARIABLE. 

The  integrals  in  the  first  members  of  these  equations  are  all 
finite  and  determinate  for  every  position  of  a  within  the  curve 
C.  Therefore  any  function  holomorphic  in  a  given  region  ad- 
mits an  infinite  number  of  successive  derivatives  at  every 
interior  point.  Each  of  these  derivatives  being  monogenic 
must  be  continuous.     Hence  the  following: 

Theorem  IV. — If/C-s^)  is  holomorphic  within  a  given  region, 
there  exists  an  infinite  number  of  successive  derivatives  of 
f{z)y  which  are  all  holomorphic  within  the  same  region. 

Denote  by  r  the  shortest  distance  from  the  point  a  to  thd 

curve  C,  Then  at  every  point  of  this  curve  |^  —  ^|  >  r.  Let 
M  be  the  upper  bound  of  the  modulus /(^)  on  Cy  and  /  the 
length  of  C,     Then 


n       fU\  ^    r    M    .     ^   Ml 

and  consequently  mod/^"^  {a)  ^ 


I .2  ... n      Ml 


27r  r"-^* 

In  particular,  if  C  is  a  circle  having  a  for  its  center, 

mod  /<«)  {a)  < . 


Art.  20.    Taylor's  Series. 

Theorem. — Let  f{z)  be  holomorphic  in  a  region  5,  and  let 
C  be  any  circle  situated  in  the  interior  of  5. 
If  a  be  the  center  and  a-\- 1  any  other  point 
interior  to  Cy 

fia  +  0  =Aci)  +  tf'{d)  +  /-/"(«)  + .  .  . 

1.2 


Taylor's  series.  45 

From  the  preceding  article,  denoting  a  variable  point  on  C 
byC, 

_  I    fAQdQr        t  I     r     , ^+'    ^     1 

=  A")  +  tna)  +  /-/"(«)  + .  .  .  +  .    r       /'"('^)  +  ^, 


where 

,  '^-  27tiJ^(Z-ar^\Z-a-t)  ^- 

By  taking  «  sufficiently  great  the  modulus  of  R  may  be 
made  less  than  any  given  positive  quantity.  Let  M  be  the 
upper  bound  of  the  modulus  of  f{s)  on  the  circle  Cy  p  the 
modulus  of  /,  and  r  the  modulus  of  C  —  <^  or  radius  of  C.    Then 

^^Jc      r''^\r-p)     ^r  —  p\rl      ' 

which,  since  p  <  r,  has  zero  for  its  limit  when  ^  =  00. 
Writing  now  z  for  a-\-t,  Taylor's  Series  becomes 

The  series  is  convergent  and  the  equality  is  maintained  for 
every  point  ^  included  within  a  circle  described  about  <3:  as  a 
center  with  a  radius  less  than  the  distance  from  a  to  the  nearest 
critical  point  oi  f{z). 

When  a  is  equal  to  zero,  Taylor's  Series  takes  the  form 
/W  =  /(o)  +  Bf'(0)  +  f-f"(o)  +  .  .  .  +  -^f'\0)  +  .  .  .  , 

1.2  1.2..  .72 

expressing /(^r)  in  terms  of  powers  of -s".     This  form  is  known 
as  Maclaurin's  Series. 


46  functions  of  a  complex  variable. 

Art.  21.    Laurent's  Series. 

Theorem. — Let  S,  a  portion  of  the  ^--plane  bounded  by  two 
concentric  circles  C,  and  C^,  be  situated  in  the  interior  of  the 
region  £f  in  which  a  given  function  /{js)  is  holomorphic.  If  a 
be  the  common  center  of  the  two  circles,  and  a  -\-  t  a.  point 
interior  to  5,  /{a  +  t)  can  be  expressed  in  a 
convergent  double  series  of  the  form 

tn  =  oo 
m  =  —  00 

With  a  -{-t  diS  di  center  construct  a  circle 
c  sufficiently  small  to  be  contained  within 
the  region  5.  If  then  6',  be  the  greater  of 
the  two  given  circles,  it  follows  from  Article  i8  that 

27ti*^Cx  Q  —  a  —  t       27ti*^c^  C,  —  a  —  t       2ni  ^cZ  —  ^  —  ^ 
But  from  Article  19, 

whence 

•^^    ^^       2ni^c^C,  -  a  -  t        2ni^c,c-a  -  t 
The  two  integrals  of  the  right-hand  member  may  be  written  : 

where 

I    r        t-^'AOdz 
^'  ~  27ii^^^{Z  -  ^)^+xc  -a-ty 

_i_  /-(C  -  ^)-^7(CKC 

But  |/|  <|C  — ^1    at  every  point  of  C,,  and  |/|>|C  — «|  at 
every  point  of  (7,,  so  that  i?,  and  R^  both  have  zero  for  a  limit 


LAURENT'S    SERIES.  47 

when  «  =  00 .  The  value  oif{a  -\- 1)  can  therefore  be  expressed 
in  the  form 

Since  in  the  region  5  the  function  f{z)/{z  —  <3:)'"+*  is  holomor- 
phic  for  both  positive  and  negative  values  of  m^  A^.  maybe 
written 

where  C  is  any  circle  concentric  with  C^  and  C^  and  included 
between  them. 

The  series  thus  obtained  is  convergent  at  every  point  a-\-t 
contained  within  the  region  S.  It  is  important  to  notice,  how- 
ever, that  when  the  positive  and  negative  powers  of  /  are  con- 
sidered separately,  the  two  resulting  series  have  different 
regions  of  convergence.  The  series  containing  the  positive 
powers  of  /  converges  over  the  whole  interior  of  the  circle  C^  ; 
while  the  series  of  negative  powers  of  /  converges  at  every 
point  exterior  to  the  circle  C^.  The  region  5"  can  be  regarded, 
therefore,  as  resulting  from  an  overlapping  of  two  other 
regions  in  which  different  parts  of  Laurent's  Series  converge. 

Writing  z  ior  a  -\-  t,  Laurent's  Series  takes  the  form 

f{z)  =  A,  +  Aiz  -  a)+A,{z  -  ay  +  .  .  , 

+  A  ^,  (z-  a)-'  +  A_,(z  -  a)-'  +  .  . . 
Consider  as  a  special  numerical  example  the  fraction 

^ = L_ L_  .  __J__ 

(<3:  -    I)   (^  -  2)   (^-  3)         2(Z-I)  Z  —  2^2{Z~2,) 

If  1^1  <  I,  all  three  terms  of  the  second  member,  when 
developed  in  powers  of  z,  give  only  positive  powers.  If 
I  <  1^1  <  2,  the  first  term  of  the  second  member  gives  a  series 
of  negative  descending  powers^  but  the  others  give  the  same 
series  as  before.  If  2  <|^|<  3,  the  first  and  second  terms 
both  give  negative  powers.     If  \z\  >  3,  all  three  terms  give 


48  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

negative  powers,  and  the  development  of  the  given  fraction 
can  contain  no  positive  powers.  Thus  a  system  of  concentric 
annular  regions  is  obtained  in  each  of  which  the  given  frac- 
tion is  expressed  by  a  convergent  power-series.  Laurent's 
Series  gives  analogous  results  for  every  function  which  is  holo- 
morphic  except  at  isolated  points  of  the  ^-plane. 

Art.  22.    Fourier's  Series. 

Let  w  =■ /{z)  be  holomorphic  in  a  region  S^y  and  let  it  be 
periodic,  having  a  period  equal  to  oo,  so  that^<8:+  noo)  =  f{2)r 
where  n  is  any  positive  or  negative  integer.  Denote  by  5„  the 
region  obtained  from  S^  by  the  addition  of  noo  to  2 ;  and  sup- 
pose that  the  regions  .  .  . ,  5_„ ,  .  .  . ,  5. 1 ,  5^ ,  5, ,  .  .  . ,  5„  ,  ... 
meet  or  overlap  in  such  a  manner  as  to  form  a  continuous  strip 
Sy  in  which,  of  course,  the  function  w  will  be  holomorphic. 
Draw  two  parallel  straight  lines,  inclined  to  the  axis  of  reals  at 
an  angle  equal  to  the  argument  of  oo,  and  contained  within  the 
strip  5.  The  band  T  included  between  these  parallels  will  be 
wholly  interior  to  S. 

2itiz 

By  means  of  the  transformation  z'  =  e  '^  the  band  T  in 
the  ^-plane  becomes  in  the  ^'-plane  a  ring  T'  bounded  by  two 
concentric  circles  described  about  the  origin  as  a  center,  z  and 
z  +  noo  falling  at  the  same  point  z\  Since  w  is  holomorphic 
in  a  region  including  T,  and 

dw      dw  dz         00     _  vtx»  dw 
dz'  ~  dz  dz'  ~~  2ni     •    '^    dz' 

w  regarded  as  a  function  of  z'  will  be  holomorphic  in  T', 
Hence,  by  Laurent's  Theorem, 

w  =  ^2   A^z'^, 

»«= -00 

the  quantity  a  in  the  general  formula  of  the  preceding  article 
being  in  this  case  equal  to  zero.  Substituting  for  z'  its  value, 
the  preceding  equation  becomes 


UNIFORM    CONVERGEJS'CB.  41) 

where 

In  the  latter  integral  the  path  is  rectilinear.  Denoting  its 
independent  variable  by  C  for  the  purpose  of  avoiding  confu- 
sion, the  value  of  w  becomes 


»«=  -00 


»t  =  l  * 


Art.  23.    Uniform  Convergence. 

Let  the  series  W :=  ^o  +  ^i  +  ^a  +•  •  •  +  ^n  +  .  •  . ,  each 
term  of  which  is  a  function  of  z^  be  convergent  at  every  point 
of  a  given  region  S.  Denote  by  Wn  the  sum  of  the  first  n 
terms  of  W.  If  it  is  possible,  whatever  the  value  of  the  posi- 
tive quantity  e,  to  determine  an  integer  p,  such  that  whenever 
n>  p 

\W-  ^n|<e 

at  every  point  of  S,  the  series  W\?,  said  to  be  uniformly  con- 
vergent in  the  region  5. 

For  convergent  series  in  general  the  determination  of  p 
will  depend  on  the  value  of  z.  In  the  case  of  uniformly 
convergent  series  /  can  be  determined  simultaneously  for  all 
points  in  the  region  5. 

Uniformly  convergent  series  can  in  many  respects  be  treated 
in  exactly  the  same  manner  as  sums  containing  a  finite  num- 
ber of  terms. 


60  FUNCTIONS  OF  A   COMPLEX   VARIABLE. 

Theorem  I. — If  in  a  region  S  a  series  of  continuous  functions 

is  uniformly  convergent,  the  sum  of  the  series  is  a  continuous 
function  of  z. 

For  at  any  point  z,  W  may  be  written  in  the  form  W  =  W^-{-R\ 
and  at  a  neighboring  point  z\  W  =  W ^  -\-R!,    Hence 

n  n 

and 

\w-w\<\w^-wj\+\R\-\-m. 

But  by  choosing  n  sufficiently  great,  \R\  and  \R'\  may  both  be  made 
less  than  any  given  positive  quantity  e/3  for  all  values  of  z  and  z' 
in  S.  Having  chosen  n  thus,  W ^  becomes  the  sum  of  a  finite 
number  of  continuous  functions.  It  is  then  continuous,  and,  by 
making  \z—z'\  less  than  a  suitable  quantity  d,  "^ —  W'\  may  be 
made  less  than  e/3.     But  under  these  suppositions 

\W-W'\<e, 

W  is  therefore  continuous  at  the  point  0. 

Theorem  II. — If  in  a  region  S  a  series  of  continuous  functions 

is  uniformly  convergent,  the  integral  of  the  series,  for  any  finite 
path  L  in  the  region,  is  the  sum  of  the  integrals  of  its  terms : 

J  Wdz  =  J  w^dz^J  Widz+, . .  +J  'w^dz+. . . 

For,  writing  W  =  W^+R,  it  is  possible  to  choose  n  so  that, 
however  small  e  may  be,  ]R\<£  at  every  point  of  Z.  If  n  be  so 
chosen. 


fwdz  ^  f^VJz  +  fRdz. 


UNIFORM   CONVERGENCE.  61 

But,  by  Article  15,  denoting  by  /  the  length  of  the  path  L, 

mod  /  Rdz<elj 
which,  when  w=  00,  has  zero  for  its  limit.    Hence 
fwdz=  lim  Jwjz. 

L  n  =  oo      L 

From  the  preceding  demonstration  we  have  at  once  the  following 
result : 

Theorem  III. — If  in  a  simply  connected  finite  region  S  a  uni- 
formly convergent  series  of  holomorphic  functions  is  integrated 
term  by  term,  the  resulting  series  is  uniformly  convergent  in  the 
same  region. 

For  in  a  simply  connected  region  the  integral  of  a  holomorphic 
function  is  independent  of  the  form  of  the  path  of  integration. 
Only  paths  whose  lengths  have  a  finite  upper  bound  need,  there- 
fore, be  considered. 

Theorem  IV. — If,  in  a  region  5,  the  series  of  uniform  functions 

is  convergent,  and  the  series 

dz      az  az 

is  uniformly  convergent,  and  if  further  the  terms  of  PT'  are  con- 
tinuous in  the  same  region,  W  will  be  the  derivative  of  W. 

For,  integrating  W^  from  a  to  z  along  a  path  L  contained  in 
S,  we  have,  by  Theorem  II, 

J  W'dz='w^(z)-w,(a)+, .  .+w^(z)-w^(a)+... 
=  W{z)-W{a). 

But  since  W  is  continuous,  it  is  the  derivative  of  the  first  mem- 
ber, and  therefore  of  the  second  member,  and  of  the  function  W, 


52  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

Theorem  V. — If  in  a  finite  region  S  the  terms  of  a  uniformly 
convergent  series 

are  holomorphic,  the  sum  of  the  series  is  holomorphic,  its  deriva- 
tive being  the  sum  of  the  derivatives  of  its  terms. 

For  let  C  be  the  boundary  of  5,  and  let  C  be  a  closed  curve 
interior  to  C.  Let  ^  be  a  positive  number  such  that  the  distance 
between  C  and  C  is  everywhere  greater  than  d.  Then  if  z  is  any 
point  interior  to  C,  we  will  have,  when  (^  varies  along  C, 

\t:-z\>3. 

The  given  series  being  uniformly  convergent,  we  can  write 

W=-W„-\-R, 

where  \R\<s  when  n  is  taken  sufficiently  great.     Accordingly  if 
L  be  the  length  of  C,  we  will  have  in  the  equation 

the  modulus  of  the  last  term  less  than 


It  follows  that  the  series 


converges  uniformly.     But  this  gives  at  once,  if  we  divide  by  2^i, 
From  the  preceding  demonstration  we  have  at  once; 


UNIFORM   CONVERGENCE.  53 

Theorem  VI. — If  a  series  of  holomorphic  functions  is  uni- 
formly convergent  in  a  given  region  S,  the  series  formed  by  the 
derivatives  of  its  terms  will  be  uniformly  convergent  in  the  same 
region. 

To  illustrate  by  an  example  that  uniformity  of  convergence 
is  essential  to  the  preceding  theorems,  take  the  series 


W  =  ^-+I 


I+Z        ,(l+Z^)(l+2^+0* 

At  the  point  z=i  each  term  is  continuous,  and  the  series 
is  convergent,  having  the  value  1/2.  The  series  is,  however, 
discontinuous  at  z=i.     For,  writing  it  in  the  form 


the  sum  of  the  first  n  terms  is  seen  to  be 

W  =^~. 

/ 

But  W  is  the  limit  of  TF^  when  w=oo,  and  is  therefore 
unity  at  every  point  z  for  which  lzl<i,  and  zero  at  every  point 
for  which  l^l  >i. 

If  now  this  series  be  considered  for  the   points  within   and 

upon  a  circle  described  about  the  origin   as   a   center  with  an 

assigned  radius  less  than  unity,  the  remainder  after  n  terms,  or 

z" 
I— pr„=-— — -  can,  by  a  suitable  choice  of  n,  be  made  less  in 

absolute  value  than  any  given  quantity.  In  such  a  region,  then, 
the  series  converges  uniformly,  and,  by  Theorem  I,  can  have  no 
point  of  discontinuity.  A  similar  result  holds  for  the  region 
exterior  to  any  circle  described  about  the  origin  as  a  center  with 
an  assigned  radius  greater  than  unity. 


54  FUNCTIONS   OF  A   COMPLEX   VARIABLE. 


Art.  24.    Power  Series. 

The  nost  elementary  and  at  the  same  time  the  most  impor- 
tant series  of  functions  which  enters  into  the  theory  of  functions 
is  of  the  form 

where  a^^  a^,  a^, .  .  .  ,  a^, .  .  .  are  constants. 

If  this  series  is  convergent  for  a  certain  value  Z  of  the  varia- 
ble 2,  it  will  be  convergent  for  every  value  of  z  for  which  \z\  <  \Z\, 
For  if  the  modulus  of  z  is  less  than  that  of  Z,  the  series 

z      z^  2** 

is  an  absolutely  convergent  geometrical  progression.  Since,  now, 
the  series 

a^-^a,Z-^a^Z''+. .  .+a^Z"  +  .  .  . 

is  convergent,  the  moduli  of  its  terms  must  have  a  finite  upper 
bound  A.  We  can  accordingly  use  its  terms  as  multipliers  foi 
the  corresponding  terms  of  the  geometrical  progression,  and  we 
will  obtain  an  absolutely  convergent  series.  But  this  series  will  be 
the  given  series 

a^^a^z-\-a^z^^-.  .  .+fl^z"  +  .  .  . 

subject  only  to  the  condition  that  |z|<|Z|. 

It  is  obvious  that  every  power  series  of  the  form  here  given 
converges  for  z  =  o.  When  we  consider  other  values  of  z  three 
cases  arise: 

(i)  The  series  may  converge  for  every  finite  value  of  z,  as, 
for  example, 

z^  z** 

I+Z  +  -  +  ...  + +... 

2  I  .  2  .  .  .  W 


POWER   SERIES.  55 

(2)  The  series  may  diverge  for  every  value  of  z,  except  z=o, 
as,  for  example, 

1+Z+42H. .  .+w**2**+. .. 

(3)  The  series  may  converge  for  some  values  of  z  different 
from  zero  and  diverge  for  others.     For  example,  the  series 


z    z^  z^ 

12  n 


converges  for  3=  —  i  and  diverges  for  z=i. 

In  the  third  case  the  modulus  of  the  values  of  z  for  which  the 
series  converges  must  have  a  finite  upper  bound.  Call  this  R. 
The  circle  of  radius  R  described  about  the  origin  as  a  center  is 
known  as  the  circle  of  convergence.  For  this  circle  we  have  the 
following  theorem: 

Theorem. — A  power  series  is  convergent  at  every  point  inte- 
rior to  its  circle  of  convergence,  and  is  divergent  at  every  point 
exterior  to  its  circle  of  Convergence. 

No  general  statement  can  be  made  as  to  the  convergence  or 
divergence  of  the  series  upon  the  circumference  of  the  circle  of 
convergence.  The  series  may  converge  at  all  points  of  the  cir- 
cumference, as,  for  example. 


2^  2** 

I+2+-2  +  ...  +  -2  +  . 


or  it  may  diverge  at  all  such  points,  as,  for  example, 


1+Z+22H. .  .+wz«+. . . , 

or  finally,  as  already  illustrated,  it  may  converge  at  some  points 
and  diverge  at  other  points  of  this  circumference. 


56  FUNCTIONS  OF  A   COMPLEX   VARIABLE. 


Art.  25.    Uniform  Convergence  of  Power  Series. 

Theorem  I. — A  power  series  is  uniformly  convergent  in  every 
circle  described  about  the  origin  as  a  center  with  a  radius  less 
than  R.     For,  if  R'  <  R,  the  series 

K|+K|i?'+...+K|i?'-+... 

is  convergent;  and,  consequently,  whatever  the  value  of  the  posi* 
tive  quantity  £,  we  can  find  an  integer  p  such  that  \in>p 

For  all  values  of  z  within  the  circle  of  radius  R',  the  sum  of 
the  series  will  then  differ  from  the  sum  of  its  first  n  terms  by  a 
quantity  less  than  £  in  absolute  value.  Hence  the  series  is  uni- 
formly convergent  within  the  circle  of  radius  R'. 

Theorem  II. — If  a  power  series  is  uniformly  convergent  in  a 
given  circle,  the  series  obtained  by  integrating  its  terms  or  by 
differentiating  its  terms  is  uniformly  convergent  in  the  same 
circle. 

This  theorem  follows  at  once  from  Theorems  III  and  VI  of 
Article  23.  Since  R  is  the  upper  bound  of  R\  the  series  of  primi- 
tives and  the  series  of  derivatives  have  exactly  the  same  circle  of 
convergence  as  the  given  po\ver  series.  We  have  also  as  an  im- 
mediate consequence  of  Theorems  II  and  V  of 'Article  23: 

Theorem  III. — The  primitive  of  a  power  series  is  the  sum  of 
the  primitives  of  its  terms;  and  the  derivative  of  a  power  series 
is  the  sum  of  the  derivatives  of  its  terms. 

As  a  result  of  these  theorems,  we  have  that,  so  far  as  continuity, 
differcntiabihty,  and  integrabilily  are  concerned,  a  power  series 
has  within  its  circle  of  convergence  the  same  properties  as  the 
simi  of  a  finite  number  of  powers. 


UKIFORM   FUNCTIONS  WITH   SINGULAR  POINTS.  57 


Art.  26.     Uniform  Functions  with  Singular  Points. 

Theorem  I. — A  function  holpmorphic  in  a  region  6"  and 
not  equal  to  a  constant,  can  take  the  same  value  only  at  iso- 
lated points  of  5. 

For  in  the  neighborhood  of  any  point  a  interior  to  5,  by 
Taylor's  theorem, 

Az)  -M  =  {z-  aY'{a)  +  (l=|)!  /"(«)  +  .  . . 

Unless  y(^)  is  constant  over  the  entire  circle  of  convergence  of 
this  series,  the  derivatives  /'{a)^  f"{d),  .  .  .  cannot  all  be 
equal  to  zero.  Let/"^''^(^)  be  the  first  which  is  not  equal  to 
zero.     Then 

f{z)  -  f{a)  =  (z-ay[-J^-M 1 f^^'^'Y)       J^-a)+ . .  .1 

Since  the  series  within  the  brackets  represents  a  contin- 
uous function,  if  \z  —  a\  be  given  a  finite  value  sufificiently 
small,  the  modulus  of  the  first  term  of  the  series  will  ex- 
ceed the  sum  of  the  moduli  of  all  the  other  terms,  and  the 
same  result  will  hold  for  every  still  smaller  value  of  \z—a\. 
For  values  of  z,  then,  distant  from  a  by  less  than  a  certain 
finite  amount, /(^r)  —  fia)  is  different  from  zero. 

If,  on  the  other  hand,  the  function  is  constant  over  the  en- 
tire circle,  described  about  ^  as  a  center,  within  which  Taylor's 
series  converges,  it  will  be  possible,  by  giving  in  succession 
new  positions  to  the  point  a,  to  show  that  the  value  of  the 
function  is  constant  over  the  whole  region  5. 

Theorem  II. — Two  functions  which  are  both  holomorphic 
in  a  given  region  5  and  are  equal  to  each  other  for  a  system  of 
points  which  are  not  isolated  from  one  another,  are  equal  to 
each  other  at  every  point  of  S. 

For  let  f{z)  and  <i){z)  be  two  such  functions.  By  the  pre- 
ceding theorem,  the  difference /(<s:)  —  0(^)  must  be  equal  to 
zero  at  every  point  of  5. 


58  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

Theorem  III. — A  function  which  is  holomorphic  in  every 
part  of  the  <s:-plane,  even  at  infinity,  is  constant. 

For,  a  being  any  given  point,  whatever  the  value  of  z^ 

Az)  =  Aa)  +  (^-  aV'{a)  +  ...  +  Y^fr^„f"V)  +  •■■ 

But  by  Article  19,  r  being  the  radius  of  any  arbitrary  circle 
having  its  center  at  ^,  and  M  being  the  upper  bound  of  the 
modulus  oi  f{z)  on  the  circumference  of  this  circle, 

J    ^(n)r   \  =1.2..   .  tlM 

But  M  is  always  finite,  and  r  may  be  made  indefinitely  great, 
Hencey^'*^(.3:)  =  o  for  all  values  of  «,  and 

A^) =/{")■ 

Theorem  IV. — If  a,function/(^),  holomorphic  in  a  region  5, 
is  equal  to  zero  at  the  point  a  situated  within  S,  the  function 
can  be  expressed  in  the  form 

/(^)  =  {z-  ar,p{B), 
where  m  is  a  positive  integer,  and  (p{z)  is  holomorphic  in  5 and 
different  from  zero  at  a. 

For  in  the  neighborhood  of  the  point  a,  by  Taylor's  Theorem, 
/(z)=/{a)  +  {z-ay{a)+.., 
LetyM  (^)  be  ^he  first  of  the  successive  derivatives  at  ^  which 
is  not  equal  to  zero.     Then 

■^^  ^  ^  Li '2.  .  ,m   ^   1.2  .  .  .  {m+ ly        ^   '  J* 

which  is  the  required  form.  The  point  a  is  a,  zero  oi  f{z),  and 
m  is  its  order. 

Theorem  V. — If  the  point  ^  is  a  singular  point  of  a  given 
function /(-s"),  but  is  interior  to  a  region  5,  in  which  the  recip- 
rocal oi  f{z)  is  holomorphic,  the  function  can  be  expressed  in 
the  form 


{z  -  aY  ' 

where  m  is  a  positive  integer,  and  x{^)  is  holomorphic  in  the 
neighborhood  of  a. 


UNIFOKM   FUNCTIONS  WITH   SINGULAR   POINTS.  59 

For  by  the  preceding  theorem 

y^  =  (^  -  ^)"0(^)» 

where  (J){z)  is  holomorphic  and  not  equal  to  zero  at  z-=.  a. 
Hence 

f{z\  =       ^  ^      -      ^(^^ 

•/^^       {2-a)^'(p{2)         {z-aY 

Further,  since  in   a  region   of  finite  extent  including  the 

point  a 

X(z)  =  A,-\A,{z  -  ^)  +  ..., 

-^     '        {z—aY~         ^z  —  a^^^*' 

a  being  an  ordinary  point  for  ^(-s"). 

The  point  «  is  a  pole  oi  f{z)  and  m  is  its  order. 

Theorem  VI. — A  function,  not  constant  in  value,  and  hav- 
ing no  finite  singular  points  except  poles,  must  take  values 
arbitrarily  near  to  every  assignable  value. 

For  suppose  that  f{z)  is  such  a  function,  but  that  it  takes 
no  value  for  which  the  modulus  o{  f{z)  —  A\.s  less  than  a  given 
positive  quantity  e.     Then  the  function 

f(?)  -  A 
will  be  holomorphic  in  every  part   of  the  ^-plane,  which,  by 
Theorem  III,  is  impossible  unless /(^)  is  a  constant. 

Theorem  VII. — A  function  /(z),  having  no  singular  point 
except  a  pole  at  infinity,  is  a  rational  entire  function  of  z. 

For  the  only  singular  point  of /(  —  J  is  a  pole  at  the  origin. 

Hence 


/©= 


^+...+^+*(.), 


where  (f){z)  is  holomorphic  over  the  entire  plane,  including  the 
point  at  infinity.  (p{z)  is  consequently  equal  to  a  constant  A^, 
The  given  function  therefore  can  be  written  in  the  form 


60  FUNCTIONS   OF   A    COMPLEX    VARIABLE. 

Theorem  VIII. — A  function /(z)  whose  only  singular  points 
are  poles  is  a  rational  function  of  z. 

The  poles  must  be  at  determinate  distances  from  one  an« 
other  ;  otherwise  the  reciprocal  of  f{2)  would  be  equal  to  zero 
for  points  not  isolated  from  one  another.  The  number  of  poles 
cannot  increase  indefinitely  as  \z\  is  increased;  for  then  the 

reciprocal  of  /  (  - )  would  have  an  infinite  number  of  zeros  indefi- 
nitely near  to  the  origin.  The  total  number  of  poles  is  there- 
fore finite.  Let  a^  b,  ,  ,  .  denote  them.  In  the  neighborhood 
of  a  the  function  can  be  expressed  in  the  form 

a  being  an  ordinary  point  for  0(^).  In  the  neighborhood  of  by 
(p{2)  can  be  expressed  in  the  form 

a  and  b  being  both  ordinary  points  for  tp{2).     Proceeding  in 

this  way  the  given  function  will  be  expressed  as  the  sum  of  a 

finite  number  of  rational  fractions  and  a  term  which  can  have 

no  singular  point  except  a  pole  at  infinity.      This  term  is  a 

rational  entire  function. 

Theorem  IX. — If  the  function  /(z)  has  no  zeros  and  no 

singular   points  for  finite  values  of  z,  it  can  be  expressed  in 

the  form  /(z)  =  e^^^\  where  £^{z)  is  holomorphic  in  every  finite 

region  of  the  z-plane. 

f(z)    ■ 
For'^-Ap  can  have  no  singular  points  except  at  infinity,  since 

/(2) 

in  every  finite  region  of  the  ^-plane  /{z)  and  /'(z)  are  holomor- 
phic and  /(z)  is  different  from  zero.  Hence,  choosing  an  arbi- 
trary lower  limit  z^y  the  integral 

is  holomorphic  in  every  finite  region.  The  function  f{z)  con- 
sequently must  take  the  form 


RESIDUES.  61 

where  g{z)  =  h{z)  +  log/CzJ. 

Theorem  X. — If  two  functions /(z)  and  0(z)  have  no  singu- 
lar points  in  the  finite  portion  of  the  z-plane  except  poles,  and 
if  these  poles  are  identical  in  position  and  in  order  for  the  two 
functions,  and  their  zeros  are  also  identical  in  position  and 
order,  there  must  exist  a  relation  of  the  form 

f{z)  =  0(^)^(^), 

where  g{j3)  is  holomorphic  in  every  finite  region  of  the  ^-plane. 

For  the  ratio  of  the  two  functions  has  no  zeros  and  no 
singular  points  in  the  finite  portion  of  the  2-plane. 

Art.  27.     Residues. 

If  a  uniform  function  has  an  isolated  singular  point  a,  it 
is  expressible  by  Laurent's  series  in  the  region  comprised  be- 
tween any  two  concentric  circles  described  about  a  with  radii 
less  than  the  distance  from  a  to  the  nearest  singular  point. 
Hence  in  the  neighborhood  of  a 

The   coefficient   of  {z  —  d)'"^   in  this  expansion   is  called  the 
"residue"  of/(^)  at  the  point  a. 

If  any  closed  curve  C  including  the  point  a  be  drawn  in  the 
region  of  convergence  of  this  series,  and  f{z)  be  integrated 
along  C  in  a  positive  direction,  the  result  will  be 


L 


f{z)dz  =  27tiB^. 
c 

The  following  may  be  regarded  as  an  extension  of  Cauchy's 
theorem  : 

Theorem  I. — If  in  a  region  5  the  only  singular  points  of  the 
one-valued  function  f{z)  are  the  interior  points  ^,  «',...  ,  the 


i 


62  FUNCTIONS   OF- A   COMPLEX   VARIABLE. 

integral  J  f(z)dz  taken  around  its  boundary  C  in  a  positive 
direction  is  equal  to 

where  B,  B' ,  ,  ,  .  are  the  residues  of  f{z)  at  the  singular  points. 
For  the  integral  taken  along  C  is  equal  to  the  sum  of  the 
integrals  whose  paths  are  mutually  exterior  small  circles  de- 
scribed about  the  points  a^  a\  ,  ,  , 

The  following  theorems  are  immediate  consequences  of  the 
preceding : 

Theorem  II. — If  in  a  region  having  a  given  boundary  Cthe 
only  singular  points  of  the  one-valued  function  f(z)  are  poles 
interior  to  C,  an  equation 

exists,  M  denoting  the  number  of  zeros  and  N  the  number  of 
poles  within  C,  each  such  point  being  taken  a  number  of  times 
equal  to  its  order. 

For  in  the  neighborhood  of  the  point  a 

f{z)  =  {z-  ar<p{z) 

where  (p{z)  is  finite  and  different  from  zero  at  a^  and  m  is  a, 

positive  integer  if  <a:  is  a  zero,  a  negative  integer  if  ^  is  a  pole. 

Hence 

/\z)  __     m  (l>\z) 

f(z)       z-a^  <t>(zy 

The  integrand,  therefore,  has  a  pole  at  every  zero  and  pole  of 
/(^),  and  its  residue  is  the  order,  taken  positively  for  a  zero, 
and  negatively  for  a  pole. 

Theorem  III. — Every  algebraic  equation  of  degree  n  has  n 
roots. 

For  let  f{z)  represent  the  first  member  of  the  equation 
^«  _|_  ^j^~-'  -|_  .  .  .  _|-  ^^  =  o.     Since  f{z)  has  no  poles  in  the 


IXTEGRxVL    OF    A    UNIFORM    FUXCTIOIN".  63 

finite  part  of  the  ^-plane,  the  number  of  roots  contained  within 
any  closed  curve  C  will  be  given  by  the  integral 


But  taking  for  C  a  circle  described  about  the  origin  as  a 
center  with  a  very  great  radius,  this  integral  is 

j_    rn.-'  +  (n-,)a,^-'  +  ...  ^^  ^  j_  fndz 

where  e  has  zero  for  a  limit  when  |^|=  oo .     Hence  the  limit 
of  the  preceding  integral,  as  \z\  is  increased,  is  n. 

Prob.  17.  Show  that  if  2  =  00  is  an  ordinary  point  of/(s^),  that 
is,  if /(^)  is  expressible  for  very  great  value  of  2  by  a  series  contain- 
ing only  negative  powers  of  z,  the  integral  of/(2;)  around  an  infinitely 

great  circle  is  equal  to  27ti  into  the  coefficient  of  — .     This  coeffi- 

z 

cient  with  its  sign  changed  is  called  the  residue  for  2  =  00  . 

Prob.  18.  Show  that  the  sum  of  all  the  residues  oi  f{z),  of  the 
preceding  problem,  including  the  residue  at  infinity,  is  equal  to 
zero. 

Prob.  10.   If  -77-^  is  a  rational  function  of  which  the  numerator 
tp{z) 

is    of   degree  lower  by  2  than  the  denominator,  and  if  the  zeros 

^j!, ,  ^, ,  .  .  . ,  ^„  of  the  denominator  are  of  the  first  order,  show  that 


Art.  28.     Integral  of  a  Uniform  Function. 

It  was  shown  in  Article  18  that,  if  a  function /"(-s-)  is  holo- 
morphic  in  a  simply  connected  region  S,  its  integral  taken 
from  a  fixed  lower  limit  contained  in  5  to  a  variable  upper 
limit  ^  is  a  uniform  function  of  z  within  5".  If  F{z)  is  a  function 
which  takes  a  determinate  value  F{zq)  at  2-  =  ^y  and  is  uniform 
while  2  remain^  within  S,  having  at  every  point  f{z)  for  its 
derivative,  the  integral  of  f{z)  from  z^  to  z  is  equal  to 
F{z)  —  F(2^).     If  F^(^z)  is  another  function  fulfilling  these  con- 


64  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

editions,  so  that  the  integral  of  f{z)  can  be  written  also  in  the 
form  FJ^z)  —  F^(z^),  the  functions  F(z)  and  FXz)  differ  only  by 
a  constant  term  ;  for 

Suppose  now  thaty"(^)  is  still  uniform  in  5,  but  that  it 
has  isolated  critical  points  ^, ,  ^,,  .  .  .  interior  to  5.  Any  two 
paths  from  z^  to  Zy  which  inclose  between  them  a  region  con- 
taining none  of  the  points  ^,,  <3:,,  . . .,  will  give  integrals  identi- 
cal in  value.  Let  the  two  paths  Z, ,  L  include  between  them 
a  single  critical  point  a^\  and  consider  the  integrals  along 
these  two  paths.  The  integral  along  Z,  will  be  equal  to  the 
integral  along  the  composite  path  Z,Z~'Z,  where  the  exponent 
—  I  indicates  that  the  corresponding  path  is  reversed  ;  for  the 
integral  along  L'^L  is  equal  to  zero.  But  L^L~^  is  a  closed 
curve,  or"  loop,"  including  the  critical  point  a^y  and,  assuming 
that  it  is  described  in  a  positive  direction  about  a^,  the  inte- 
gral along  it  is  equal  to  27iiB^y  where  B^  is  the  residue  oi  f{z) 
at  a^.     Hence 

rf{z)dz  =  27iiB^  +  ff{z)dz. 

If  now  the  two  paths  Zj,  L  from  z^  to  z  include  between 
them  several  critical  points  a^,  a^,  «^,  .  .  .,  draw  intermediate 
paths  Z-„  .  .  .,  L^,  so  that  the  region  between  any  two  consec- 
utive paths  contains  only  one  critical  point.  The  integral 
along  Z,  will  be  equal  to  the  integral  along  the  composite  path 
L^L'^L^  .  .  .  L^'^L^L'^L,  since  the  integrals  corresponding  to 
L'^L^,  .  .  .,  L^'^L^y  L~^L  are  all  equal  to  zero.  But  L^L'^y 
L^L^'\  .  .  .,  L^L'^  are  all  closed  paths  or  loops,  each  including 
a  single  critical  point,  so  that,  assuming  that  each  is  described 
in  a  positive  direction  and  that  B^,  B^,  ^^,  .  .  .  denote  the  resi- 
dues of  f{z)  at  the  critical  points, 

£^A^)dz  =  2ni[B^  +  ^,  +  ^^  +  .  .  .)  +  fj{z)dz. 

It  has  been  assumed  in  the  preceding  that  neither  of  the 
paths  Zj,  L  intersects  itself.     In  the  case  where  a  path,  for 


IKTEGRAL   OF   A   UNIFORM   FUNCTION^.  65 

example  Zj,  intersects  itself  in  several  points  c^,  <;,,...,  it  is 
possible  to  consider  Z,  as  made  up  of  a  path  Z/  not  intersect- 
ing itself,  together  with  a  series  of  loops  attached  to  Z/  at 
the  points  c^y  c^,  .  .  .  Each  of  these  loops  encloses  a  single 
critical  point  Uk  and,  if  described  in  a  positive  direction,  adds 
to  the  integral  a  term  27tiB^  Each  such  loop  described  in  a 
negative  direction  adds  a  term  of  the  form  —  2niB^,  It  is  evi- 
dent that  the  form  of  each  loop  and  the  point  at  which  it  is 
attached  to  Z/  may  be  altered  arbitrarily  without  altering  the 
value  of  the  integral,  provided  no  critical  point  be  introduced 
into  or  removed  from  the  loop.  In  fact  all  the  loops  may  be 
regarded  as  attached  to  Z/  at  z^. 

It  can  be  proved  by  similar  reasoning  that  the  most  gen- 
eral path  that  can  be  drawn  from  z^  to  z  will  be  equivalent,  so 
far  as  the  value  of  the  integral  is  concerned,  to  any  given  path 
Z  preceded  by  a  series  of  loops,  each  of  which  includes  a  sin- 
gle critical  point  and  is  described  in  either  a  positive  or  nega- 
tive direction.  The  value  of  the  integral  is  therefore  of  the 
form 

f{z)dz  +  27ii{m,B,  +  m^B^  +  •••)» 


I 


where  m^^  m^^  .  .  .  are  any  integers  positive  or  negative. 

/*'    dz 

As  an  example  consider  the  integral   / .      The  only 

critical  point  is  ^  =  ^.  Any  path  whatsoever  from  z^  to  z  is 
equivalent  to  a  determinate  path,  for  example,  a  rectilinear 
path,  preceded  by  a  loop  containing  a  and  described  a  certain 
number  of  times  in  a  positive  or  negative  direction.  If  w  de- 
note the  integral  for  a  selected  path,  the  general  value  of  the 
integral  will  be  w  +  '2'nni.  If  now  a  straight  line  be  drawn 
joining  z^  to  a,  and  if  along  its  prolongation  from  a  to  infinity 
the  -sr-plane  be  cut  or  divided,  the  integral  in  the  ^-plane  thus 
divided  is  one-valued.  But,  with  the  variation  of  z  thus  re- 
stricted, any  branch  of  the  function  log  {z  —  a)  is  one-valued. 
Select  that  branch,  for  example,  which  reduces  to  zero  when 
z=^  a-\-  \.     It  takes  a  determinate  value  for  z  =  z^,  and  its 


66  FUNCTIONS   OF   A    COMPLEX   VARIABLE. 

derivative  for  every  value  of  z  is .     Hence,  denoting  it 

by  Log  {z  -  a), 

/*   dz  z  —  a 

— —  =  Log  {z-a)-  Log  {z,  -a)  =  Log  ^— -. 

For  a  path  not  restricted  in  any  way,  the  value  of  the  inte- 
gral is 

z  —  a 


/dz                   z  —  a 
=  Log h  2nni  =  log 
,,z  -a           ^  z,-a^  ^ 


?o  —  ^ 


Prob.  20.  If  -rT-\  is  a  rational  function  of  z  of  which  the  numer- 
i\z) 

ator  is  of  degree  lower  by  2  than  the  denominator,  and  if  the  zeros 

/Zj,  «,,...,  d!„  of  the  denominator  be  of  the  first  order,  show  that 


t/«o  i\){z)  X  ip\av)      ^  z^  —  Uy 


where  2<p{av)/tp'(ay)  =  o.  (See  Prob.  19,  Art.  27.) 

1 


Art.  29.    Weierstrasss  Theorem. 

Any  rational  entire  function  of  z,  having  its  zeros  at  the 
points  ^,,  ^,,  .  .  .,  ^^,  can  be  put  in  the  form 

A(z-  a,p{z  -  a,y^  .  .  .  (^  -  ^^)''m, 

where  ^  is  a  constant  and  «,,  «,,  .  ,  ,,  n„  are  positive  integers. 
More  generally,  any  function  which  has  no  singular  point  in  the 
finite  portion  of  the  ^-plane  and  has  the  points  a^,  .  .  .,  a„  as 
its  zeros,  is  of  the  form 

e^'\z  -  a,p  ..,{z-  a^Ym, 
where  ^{z)  is  holomorphic  in  every  finite  region. 

The  extension  of  this  result  to  the  case  where  a  function 
without  finite  singular  points  has  an  infinite  number  of  zeros  is 
due  to  Weierstrass.  It  is  effected  by  means  of  the  following 
theorem  : 

Theorem. — Given  an  infinite  number  of  isolated  points  a^^ 


WEIERSTEASS'S  THEORBM.  67 

a^j  .  .  .,  a„,  .  .  .,  a  function  can  be  constructed  holomorphic  ex- 
cept at_  infinity  and  equal  to  zero  at  each  of  the  given  points 
only. 

For  the  given  points  can  be  taken  so  that 

k,  I  <kJ  <•••!««'<••  •> 

I  an  I  increasing  indefinitely  with  n.  Consider  the  infinite  product 

(P{s)  =  nil  _^)^^«w, 
1  ^      ii„' 

where  Pn{^)  denotes  the  rational  entire  function 
Any  factor  may  be  written  in  the  form 


But  since 


the  path  of  integration  being  arbitrary  except  that  it  avoids 
the  points  a^,  a,,  .  .  .,  the  product  may  be  expressed  as 


00  /»2 

ne'f'n^'\  in  which  tp„{z)  =  —J 


z^dz 


In  any  finite   region  of  the  z-plane  it  will  be  possible  to 

assume  that  |  z  |  ^  P  <  |^^|,  if  P  and  m  be  suitably  chosen,  since 
\ay\  increases  indefinitely  with  n.  Divide  the  product  into 
two  parts 


n  (I-  -)- 


and 


68  FUNCTIONS  OF   A   COMPLEX   VARIABLE. 

Since  when  n>m,  \aj >pj  the  integrand  of  the  exponent 

^'^  '        Jo  a.»(a„-«) 
is  holomorphic  in  the  circle  |2;|  <|0.     Accordingly,  (l>„(z)  is  in  the 
given  region  a  holomorphic  function  of  its  upper  limit. 
But  we  may  write 

m 

00 

Consider  now  the  series  1(1)  Jz).  For  the  modulus  of  each  term 
we  have 

'^"^'^'"  KHfcp7)' 

where  /  denotes  the  length  of  the  path  of  integration.  But,  if  the 
path  of  integration  be  taken  as  rectilinear,  we  will  have  l<p. 
Hence  each  term  o!  the  series  is  less  in  absolute  value  than  the 
corresponding  term  of  a  convergent  geometrical  progression  in- 
dependent of  z.  The  series  is,  accordingly,  uniformly  conver- 
gent and,  by  Theorem  V  of  Article  23,  represents  a  function  holo- 
morphic in  the  given  region.     The  exponential 

also  must  be  holomorphic.     The  other  part  of  the  product 


m 


^■(-f.) 


i7    I--  Uf«« 


containing  only  a  finite  number  of  factors  is  everywhere  holo- 
morphic, vanishing  at  all  of  the  points  a^,  a^,  .  .  .  ,  which  are 
situated  within  the  given  finite  region.  But  this  region  may  be 
extended  arbitrarily.  The  product  therefore  fulfils  the  required 
conditions. 

In  the  preceding  demonstration  it  was  tacitly  assumed  that 
none  of  the  given  points  a^,  ^2, .  .  .  was  situated  at  the  origin. 
To  introduce  a  zero  at  the  origin  it  is  necessary  merely  to  mul- 
tiply the  result  by  a  power  of  z. 

The   most   general   function  without   finite   singular   points 


WEIERSTKASS'S    THEOREK.  69 

having  its  only  zeros  at  the  given  points  a^,  a^, .  .  .,  ^«  . .  .,  can 

be  expressed  in  the  form 

where  ^^)  is  holomorphic  except  at  infinity ;  for  the  ratio  of 
any  two  functions  satisfying  the  required  conditions  is  neither 
infinite  nor  zero  at  any  finite  point. 

By  means  of  Weierstrass's  theorem  it  is  possible  to  express 
any  function,  F^s),  whose  only  finite  singular  points  are  poles  as 
the  ratio  of  two  functions  holomorphic  except  at  infinity.  For, 
construct  a  function  ^-(-s-)  having  the  poles  of  F{2)  as  its  zeros. 
The  product  F{2).  ip{s)  =  (p(z)  will  have  no  finite  singular  point. 
The  given  function  can,  therefore,  be  written 

which  is  the  required  form. 

In  applying  Weierstrass's  theorem  to  particular  examples, 
it  will  rarely  be  found  necessary  to  include  in  the  polynomial? 
Fn{^)  SO  many  terms  as  were  employed  in  the  demonstration 
given  above.  It  is  quite  sufficient,  of  course,  to  choose  these 
polynomials  in  any  way  which  will  make  the  product  converge 
for  finite  values  of  <sr  to  a  holomorphic  function.  Factors  of  the 
form  /  „  K 

\         aj 

where  /*„(^)  is  chosen  in  such  a  manner,  are  called  "  primary 
factors." 

As  an  application  of  Weierstrass's  Theorem  take  the  reso- 
lution of  sin  2  into  primary  factors.  The  zeros  of  sin  z  are  o, 
±;r,  ±2;r,  .  .  .,  ±«;r,  ....     Consider  factors  of  the  form 


\  fZTt' 


SO  that  Pn{^)  contains  only  one  term  — ,  and 


nn 


r'        zdz 


nninn  —  z) 
0         ^  ' 


70  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

00 

The  series  I(p^{z)  will  converge  uniformly  in  any  given  finite 

m 

region.     For  if  p  and  m  be  suitably  chosen  we  will  have 

|2|<^<w;r. 
Hence 

\       mn/ 

where  /  is  the  length  of  the  path  of  integration  from  the  origin 
to  the  point  z.  If  this  path  be  taken  as  rectilinear,  we  will  have 
/  <  iO  and  (lf^(z)  will  be  less  in  absolute  value  than  the  correspond- 
ing term  of  the  convergent  numerical  series 


A  similar  result  holds  for  the  series  I  ^^{z).     These  series  ac- 

—  m 

cordingly   represent   holomorphic   functions   in   any   region   for 
which  |z|  <  /O.     Hence  the  expression  sought  is 


sin  z=ze^^'^n  ( i  — —  ]en^. 
-  00  \       nnj 


the  value  w=o  being  excluded  from  the  product.     It  will  be 
shown  in  the  next  article  that  e^^^^  =  i. 

Prob.  21.  If  (jJx  and  co^  be  two  quantities  not  having  a  real  ratio, 
the  doubly  infinite  series  of  which  the  general  term  is — 


I 


mittag-leffleb's  theorem.  71 

is  absolutely  convergent  ii  p'>2.    Hence  show  that  the  product 


.(.)=.iz(i-i) 


where  co=mo)i  +  no)2,  defines  a  holomorphic  function  in  any  finite 
region  of  the  2-plane.     This  function  is  Weierstrass's  sigma  func- 
tion, and  is  the  basis  of  his  system  of  elliptic  functions. 
Prob.  22.    Show  that  the  product 


'K-i)' 


I 


2(1  +  2 


defines  a  function  holomorphic  in  every  finite  region  of  the  2-plane. 
This  function  is  the  reciprocal  of  the  gamma  function  r(z)  or,  in  the 
notation  employed  by  Gauss,  77(2— i).  It  may  also  be  defined  as 
the  limit  when  w=  00  of  the  product 

z(z+i)(z-\-2)  .  .  .  (z-^-n)   -- 

n     • 

i-2'3  .  .  .  n 

Prob.  23.     Assuming  the  relation  that 

r{i-\-z)=zr(z\ 

show    that 

I  I         sinrrz 

r(z)*r(i-2)~~S~' 

Art.  30.    Mittag-Leffler's  Theorem. 

Any  uniform  function  j(z)  with  isolated  singular  points 
ai,  a2,  .  .  .  can  be  represented  in  the  neighborhood  of  one  of 
these  points  by  Laurent's  series;   viz., 

}{z)=Ao+Ai{z-aJ+A2(z-aj2^... 
+Bi{z-aJ-^+B2{z-aJ-2+... 

Hence  /(,)=^(,)+G„(j-^), 


72  FUNCTIONS   OF   A   COMPLEX  VARIABLE. 

where  <j>(z)  is  holomorphic  in  a  region  containing  the  point  <z„, 
and  GJ-— — j  is  holomorphic  over  the  whole  plane  excluding 

the  point  a„.  If  fl„  is  a  pole  of  /(z),  GJ  —^ — )  consists  of  a  finite 
number  of  terms;  otherwise,  it  is  an  infinite  series.  If  the  num- 
ber of  singular  points  is  finite,  and  the  function  gJ )  is 

formed  at  each  such  point,  we  can  obtain  by  subtracting  the 
sum  of  these  functions  from  j(z)  a  remainder  which  has  no 
singular  point  in  the  finite  part  of  the  plane.  This  remainder 
can  therefore  be  expressed  as  a  series  of  ascending  powers  G(z) 
converging  for  every  finite  value  of  z.  The  function  /(z)  can 
accordingly  be  written  in  the  following  form: 


m=G(z)+IG„{^), 


which  is  analogous  to  the  expression  of  a  rational  function  by 
means  of  partial  fractions. 

The  extension  of  this  result  to  the  case  where  the  number 
of  singular  points  is  infinite  is  due  to  Mittag-Leffler.  Let  ai, 
^2, . . .  ,  cin, ...  be  the  singular  points  of  the  one-valued  func- 
tion /(z),  and  suppose  that 

kil<la2l<  . . .  U„|<. . . , 

\a„\  increasing  without  limit  when  n  is  increased  indefinitely. 
Let,  further,  G„(—2 — )    be  the  series  of  negative   powers  of 

z  —  a^  contained  in  the  expansion  of  /(z)  according  to  Laurent's 
series  in  the  neighborhood  of  a„. 

The  function  gJ ),  having  no  singular  pomt  except  at 

a„,  may  be  developed  by  Maclaurin's  series  in  the  form 


mittag-leffler's  theorem.  ■  73 

\z    a^/ 

and  the  series  will  converge  uniformly  within  a  circle  described 
about  the  origin  as  a  center  with  any  determinate  radius  |0„<  \aj^. 
Hence,  for  any  point  within  the  circle  [2|  =  (0„, 

Fn(z)    representing   the   first   p   terms   of   the   development   of 

GJ )  by  Maclaurin's  theorem,  and  R  the  remainder,  which 

by  a  suitable  choice  of  p  may  be  made  less  in  absolute  value 
than  any  given  quantity. 

Choose  the  positive  quantities  £i,  ^2,  •  -  -  ,  £„,...  so  that 
the  series  £i  +  £2  +  - •  •  +  ^«  +  - •  •  is  convergent.  Choose  also  in 
connection  with  each  of  the  points  <ii,  fi^2j  •  •  •  j  ^«>  •  •  • ,  a  suitable 
integer  p  such  that 

mod[Gi(^^J  -Fi(zij  <  £i,  ifl2l^^i<  |ai|; 

mod|^G2^j^j-i^2(2)J<^2,  if  \z\<P2<la2\; 
and,  in  general, 

mod[G„^^3^j  -i^n(2) J  <  c,:,  if  \z\<Pn<\an\. 
Consider  now  the  series 

in  any  finite  region  of  the  plane,  the  points  ai,  ^2, . . . ,  a^, . , . 


74  FUNCTIONS  OF  A    COMPLEX   VARIABLE. 

being  excluded.  Since  \a^\  increases  indefinitely  with  «,  it  is 
possible,  in  any  finite  region  of  the  z-plane,  to  assume  that 
kl<Pm<kml-  Separate  from  the  series  its  first  w-i  terms. 
These  terms  will  have  a  finite  sum.  The  remaining  terms  of  the 
series  taken  in  order  will  be  less  in  absolute  value  than  £^> 
£„+i, . . .  respectively,  \z\  being  less  than  the  least  of  the  quanti- 
ties p„j  Ptn+ii '  •  •     Accordingly,  the  series 


|[°-(;^J-''-«] 


is  absolutely  convergent  for  every  value  of  z  except  a^  a^, . . . , 
a^, . . .  It  is  evident,  further,  that  in  any  given  finite  region, 
from  which  the  points  a^,  aj, . . .  ,  a„, .  .  .  are  removed  by  means 
of  small  circles  described  about  them  as  centers,  the  series 
converges  uniformly.  In  such  a  region  any  term  of  the  series 
is  holomorphic;  and,  therefore,  by  Theorem  V  of  Article  23,  the 
series  defines  a  holomorphic  function. 

The  point  a^  is  an  ordinary  point  for  the  difference 

since  in  its  neighborhood  this  difference  may  be  developed  as  a 
convergent  series  containing  only  positive  powers  of  z  —  a^.  In 
the  same  way  each  of  the  points  a^,  aj, . . . ,  o„, . . .  is  an  ordinary 
point  for  the  function 


^(^)-?^"fe)-'^«4 


This  function,  therefore,  can  have  no  singular  point  except  at 
infinity,  and  must  be  expressible  as  a  series  G{z)  containing 
only  positive  powers  of  z  and  converging  uniformly  in  any 
finite  region  of  the  z-plane.  Hence  the  function  f{z)  may  be 
put  in  the  form 


mittag-leffler's  theorem. 


75 


in  which  the  character  of  each  singular  point  is  exhibited. 

As  an  appHcation  of  Mittag-Leffler's  theorem  consider  cot  2. 
Its  singular  points  are  ^  =  o,  ±  tt,  ±  2;r,  .  .  .  .  In  the  neigh- 
borhood of  ^  —  o,  cot  ^ is  holomorphic ;  and  in  the  neigh- 

borhood  of  -^  =  njt,  n  being  any  positive  or  negative  integer, 

I 

cot  z  — 


z  —  nn 


is  holomorphic.     The  series 


+  00 


z—  nn 


in  which  m  \s  an  arbitrary  positive  integer,  is  not  convergent 
for  finite  values  of  z,  even  when  |^|  <  mn.      The  series. 


.z  —  nn      nn_ 


nn(z  —  nn) 


\         nnl 


is,  however,  absolutely  convergent  at  every  point  for  which 
1^1  <  mn.     For  the  modulus  of  any  term  is  equal  to 


nn  \  rnti 


nitt 
and,  therefore,  less  than  the  corresponding  term  in  the  series 

z\ 


\         mn] 
A  similar  result  holds  for  the  series 

I 


-^  + 


—  -T 

nn       nnj 


It  is  easy  to  see  now  that  the  reasoning  employed  in  the 
demonstration  of  Mittag-Leffler's  theorem  may  be  applied  to 
show  that  the  series 


76  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

where  the  summation  does  not  include  n  =o,  defines  a  func- 
tion holomorphic  in  any  finite  region  of  the  ^-plane,  the  points 
O,  ±  ;r,  ±  2;r, . . .  being  excluded.     The  difference 


I 

cot  2 

Z 


r— ^+-1 


can  have  no  singular  point  except  at  infinity.  It  must,  there- 
fore, be  expressible  as  a  series  G{z)  of  positive  powers  of  <sr, 
having  an  infinite  circle  of  convergence.     Hence 

I       jti^  r      I  I  "1 

—  00      l—  — ' 

The  next  step  is  to  determine  G{z).  It  is  to  be  observed 
that,  if  G{z)  is  a  constant,  its  value  must  be  zero,  since 
cot  (—  2:)  =  —  cot  z.  If  G{z)  is  not  a  constant,  differentiation 
of  the  preceding  expression  for  cot  z  gives 

I  I        ^"^         I 

~  s"h?^  ^  ^'(^^  ~?  ~  ^  {z  -  nnf 

It  follows,  by  changing  z  into  z  -{-  n,  that 

G\z  +  7t)  =  G'{z), 

Hence  G'{z)  is  periodic,  having  a  period  equal  to  n ;  and  as  the 
point  z  traces  a  line  parallel  to  the  axis  of  reals,  G'{z)  passes 
again  and  again  through  the  same  range  of  values.  But  G\z)j 
being  the  derivative  of  G{z)y  is  holomorphic  for  every  finite 
value  of  z.  It  can,  therefore,  become  infinite,  if  at  all,  only 
when  the  imaginary  part  of  z  is  infinite.  If  z  be  written  in 
the  form  x-\-iy,  the  value  of  G'{z)  may  be  expressed  as 

Q'U\  =  '  ,  ^  I /    2/4cos;tr4-/sin;tr)y 

^  ^       (x  -\-  iyf   '  ^  {x-^iy—nn)^      \(cos  2;tr+^sin  2;r)— ^>/ 

When  J  =  ±  00  the  first  and  last  terms  of  the  second 
member  vanish.     In  regard  to  the  series  it  can  be  proved  that. 


mittag-leffler's  theorem.  77 

for  any  given  region  is  which  y  is  finite  and  different  from 
zero,  an  integer  r  can  be  found  such  that  the  sum  of  the  moduli 
of  those  terms  for  which  \n\>  -k  is  less  in  absolute  value  than 
any  previously  assigned  quantity  e.  As  \y\  is  increased  the 
modulus  of  each  of  these  terms  is  diminished.  The  modulus 
of  their  sum,  therefore,  cannot  exceed  e  when  7  =±00.  But 
whenj^=  ±00  the  sum  of  any  finite  number  of  terms  of  the  series 
is  zero.  Hence  the  limit  of  the  whole  series  is  zero.  G'(z)y 
therefore,  never  becomes  infinite.  Hence,  by  Theorem  HI, 
Article  26,  it  is  constant,  and  is  equal  to  zero.  It  follows  that 
G{z)  is  equal  to  zero. 

The  expression  for  cot  z  is  accordingly 


cot  ^  =  ^  +  ^[^3^  +  ^]. 


The  logarithmic  derivative  of  the  product  expression  for 
sin  Zy  given  in  the  preceding  article  as  an  example  of  Weier- 
strass's  theorem,  is 

cot  z=g'{z)^- + ^  r — ^ — h— T 

Hence  ^(^)  in  that  expression  is  a  constant.  Making  z  =  o, 
its  value  is  seen  to  be  unity. 

Prob.  24.  From  the  expression  for  cot  z  deduce  the  equation 


where  the  summation  does  not  exclude  n  =^  o. 
Prob.   25.  Show  that  the  doubly  infinite  series 

where  go=z  mco^  +  nao^ ,  defines  a  function  whose  only  finite  singular 
points  are  z  =  go.  This  function  is  Weierstrass's  ^function.  (Com- 
pare Problem  21.) 

Prob.  26. -Prove  that 


78  FUNCTIONS   OF  A    COMPLEX   VARIABLE. 

Prob.  27.  Prove  that  ^\z)  =  —  2^7—^: — y*  where  the  summa- 
tion does  not  exclude  00=0. 


Art.  31.    Singular  Lines  and  Regions. 

The  functions  whose  properties  have  been  considered  in  the 
preceding  articles  have  been  assumed  to  have  only  isolated  sin- 
gular points.  That  an  infinite  number  of  singular  points  may- 
be grouped  together  in  the  neighborhood  of  a  single  finite 
point  is  evident,  however,  from  the  consideration  of  such  ex- 
amples as 

w  =  cot-,        w  =  e^°^^^  ^. 
z 

In  the  former  an  infinite  number  of  poles  are  grouped  in  the 
neighborhood  of  the  origin.  In  the  latter  an  infinite  num- 
ber of  essential  singularities  are  situated  in  the  vicinity  of  the 
point  z  =.  a. 

It  is  easy  to  illustrate  by  an  example  the  occurrence  of  lines 
and  regions  of  discontinuity.     Take  the  series  * 

The  sum  of  its  first  n  terms  is 

I 


z       —  I 

which  converges  to  unity  if  |^|<  i,  and  to  zero  if  |-s^!>  I. 
Hence  the  circle  \z\=.  i  is  a  line  of  discontinuity  for  this 
series. 

Consider  now  any  two  regions  5,  and  5,,  the  former  situated 
within,  the  latter  without,  the  unit  circle.  Let  (p{2)  and  ip{z) 
be  two  arbitrary  functions  both  completely  defined  in  these 
regions.     The  expression 

^(z)ii{£)+<p{z)[_i-e{z)\ 

*  This  series  is  due  to  J.  Tannery.  See  Weierstrass,  Abhandlungen  aus  der 
Functionenlehre  (1886),  p.  102. 


SINGULAR   LINES   AND    REGIONS.  ^  79 

will  be  equal  to  0(z)  in  S^  and  ^(z)  in  S^.  In  regions  com- 
pletely separated  from  one  another  by  a  singular  line,  the  same 
literal  expression  may  thus  represent  entirely  independent 
functions. 

For  a  single  continuous  region,  however,  in  the  interior  of 
which  exist  only  isolated  critical  points,  the  character  of  the 
function  in  one  part  determines  its  character  in  every  other 
part.  Let  5  be  such  a  region,  and  assume  that  its  boundary  is  a 
singular  line.  In  the  neighborhood  of  any  interior  point  a,  not 
a  critical  point,  the  given  function  is  expressible  as  a  power 
series,  viz. : 

M = /(«) + (^  -  ^yx") +■■■+ -r^^/'"' w + •  •  • 

This  series  will  converge  uniformly  over  a  circle  described 
about  «  as  a  center  with  any  determinate  radius  less  than  the 
distance  from  a  to  the  nearest  singular  point.  It  serves  for  the 
calculation  of/(^)  and  all  its  successive  derivatives  at  any  point 
^  interior  to  this  circle.  From  the  preceding  power  series,  ac- 
cordingly, can  be  obtained  another 

A^)  =  Ab) + (^  -  i>V'{.b)  +  ...  +  If^yLmb) +..., 

representing  the  f{z)  within  a  circle  described  about  /^  as  a 
center.  In  general,  the  point  b  can  be  so  chosen  that  a  portion 
of  this  new  circle  will  lie  without  the  circle  of  convergence  of 
the  former  power  series.  At  any  new  point  c  within  the  circle 
whose  center  is  b,  the  value  of  the  function  and  all  its  succes- 
sive derivatives  can  be  calculated  ;  and  so,  as  before,  a  power 
series  can  be  obtained  convergent  in  a  circle  described  about  c 
as  a  center  and,  in  general,  including  points  wot  contained  in 
either  of  the  preceding  circles.  By  continuing  in  this  manner 
it  will  be  possible,  starting  from  a  given  point  a  with  the  ex- 
pression oi  f{z)  in  ascending  powers,  to  obtain  an  expression  of 
the  same  character  at  any  other  point  k  which  can  be  connected 
with  <^  by  a  continuous  line  everywhere  at  a  finite  distance 
from  the  nearest  singular  point.    It  follows  that  the  character  of 


80  FUNCTIONS   OF   A   COMPLEX   VARIABLE. 

the  function  everywhere  within  5  can  be  determined  completely 
from  its  expression  in  ascending  power  series  in  the  neighbor- 
hood of  a  single  interior  point. 

The  process  here  described,  whereby  from  a  single  ascending 
power  series  representing  a  function  in  the  neighborhood  of  a 
given  point  of  the  z-plane  one  can  derive  a  succession  of  similar 
series,  the  totality  of  which  determines  the  function  throughout  a 
connected  region  limited  only  by  the  singularities  of  the  function, 
is  known  as  the  process  of  "  analytical  continuation."  Each  of 
the  series  obtained  is  called  an  **  element "  of  the  function.  Ac- 
cording to  the  theory  of  functions  of  a  complex  variable  as  pre- 
sented by  Weierstrass,  the  infinite  number  of  elements  connected 
together  by  the  process  of  analytical  continuation  are  said  to 
constitute  the  definition  of  an  "  analytical  function." 

It  will  be  impossible  by  the  process  just  explained  to  derive 
any  information  in  regard  to  a  function  at  points  exterior  to  the 
connected  region  S  covered  by  the  circles  of  convergence  of  its 
elements.  Moreover,  as  has  been  shown  by  an  example,  an 
expression  which  gives  a  complete  definition  of  /(z)  within  S 
may  carry  with  it  the  definition  of  an  entirely  independent  func- 
tion outside  of  5. 

As  an  example  of  a  function  having  a  singular  region  con- 
sider the  function  defined  by  the  scries 

I  -f-  2^  -f-  2^*  +  2<S''  4"  .  .  .  , 

which  represents  a  function  without  singular  points  in  the 
interior  of  the  circle  j^s"!  =  I.  For  points  on  or  without  this 
circle  the  series  is  divergent ;  and,  further,  it  is  impossible  to 
obtain  from  it  an  expression  converging  when  \s\  =  i.  The 
function  thus  defined,  consequently,  exists  only  in  the  region 
interior  to  the  unit  circle.  By  changing  j3  into  1/2  a  series 
I    2   ,    2   ,    2, 

is  obtained,  representing  a  function  which  has  no  existence  in 
the  interior  of  the  unity  circle.  Functions  in  connection  with 
which  such  regions  arise  are  called  "lacunary  functions."* 

*  Poincar6,  American  Journal  of  Mathematics,  Vol.  XIV;  Harkness  and 
Morley,  Theory  of  Functions  (1893),  p.  119 


FUNCTIONS   HAYING   n.  VALUES.  81 


Art.  32.     Functions  Having  n  Values. 

Let  the  function  w  =/{^)  take  at  the  point  ^^  of  a  given  re- 
gion 5  a  value  w^^K  Suppose  that  along  any  continuous  path, 
beginning  at  :s^,  and  subject  only  to  the  conditions  that  it  shall 
remain  in  the  interior  of  5  and  shall  not.  pass  through  certain 
isolated  points  a^ ,  a^ ,  .  .  .  ,  w  is  continuous  and  has  a  contin- 
uous derivative.  If  it  is  impossible,  when  2  traces  such  a  path, 
to  return  to  the  point  -s-^  so  as  to  obtain  there  a  value  of  w  dif- 
ferent from  w^^\  w  is  uniform  in  the  region  5.  On  the  other 
hand,  certain  paths  may  lead  back  to  z^  with  new  values  of  w. 

Suppose  that  at  each  point  of  S,  except  «, ,  ^, ,  .  .  . ,  zf  has 
n  different  values,  and  that  starting  from  such  a  point  z^  and 
tracing  any  continuous  curve  not  passing  through  <?;,  ,  ^j,  .  .  .  , 
the  several  values  of  w  give  rise  to  n  branches  ze/j ,  w, ,  .  .  .  ,  w^, 
each  of  which  is  characterized  by  a  continuous  derivative.  In 
the  neighborhood  of  a^  any  one  of  the  points  a^^  a^^  ,  .  . 
these  branches  are  said  to  be  distinct  or  not,  according  as  small 
closed  curves  described  about  this  point  lead  from  each  value  of 
w  back  to  the  same  value  again,  or  cause  some  of  the  branches 
to  interchange  values.  In  the  latter  case  the  point  is  a  branch 
point. 

About  any  branch  point  Uk  as  a  center  describe  a  small  cir- 
cle ;  and  suppose  that,  starting  from  any  point  of  it  with  the 
value  Wa.  corresponding  to  a  certain  branch,  the  values 
w^  ^Wy  ...  are  obtained  by  successive  revolutions  about  ak , 
the  original  value  being  reproduced  after  p  revolutions.  In- 
troduce now  a  new  independent  variable  2'  such  that 

z'  =  {z-a,fK 

It  can  be  shown  that  when  z  makes  one  revolution  about 
ak ,  z'  makes  only  one  /th  part  of  a  revolution  about  the  ori- 
gin  of  the  .a'-plane,  and  that  to  a  complete  revolution  of   z^ 


82  FUNCTIONS   OF   A    COMPLEX   VARIABLE 

about  the  origin  of  the  ^'-plane  correspond  /  revolutions  of  z 
about  a^.  Considering  then  the  branch  w^  as  a  function  of  z' , 
the  origin  cannot  be  a  branch  point,  for  whenever  z'  describes 
a  small  circle  about  it,  the  value  w^.  is  reproduced.  The 
branch  Wa^  must  accordingly  be  expressible  by  Laurent's 
series  in  the  form  +«, 

—  00 

or,  substituting  for  z'  its  value, 

1  3 

w^  =  A,-\-  A,{z  -  a^V  +  AJ,z  -  a^*  +  . . . 

+  ^_.(-sr -^,r^  +  yi_,(^  -  ^,)"^+ .  .  . 

This  expression  makes  plain  the  relation  between  the  different 
branches  of  a  function  in  the  neighborhood  of  a  branch  point. 
When  the  development  of  a  branch  in  the  neighborhood  of  one 
of  its  branch  points  gives  rise  to  only  a  finite  number  of  terms 
containing  negative  powers,  the  branch  point  is  called  a  "  polar 
branch  point." 

Consider  the  functions 

P^   =  W^W^  +  W^W^  •\-  .  .  .-\-  Wn-^Wny 


P^   =   W^W^  .  .    .Wn. 

Each  of  these  functions  is  unchanged  in  value  when  several  or  all 
of  the  quantities  w^,  ee/,,  .  ,  *  y  w^  are  interchanged,  and  is  con- 
sequently a  one-valued  function  of  z  within  5.  Hence  w  must 
satisfy  an  equation  of  the  «th  degree, 

w^  +  P^w''-'  +  P^  tv^-'  +  .  .  .  +  p^  =  o, 
the  coefficients  of  which  are  one-valued  functions  of  z  having 
only  isolated  critical  points  within  S.  When  the  entire  ^-plane 
can  be  taken  as  the  region  S,  and  those  branch  points  at  which 
the  branches  do  not  all  remain  finite  are  polar  branch  points, 
the  only  other  critical  points  being  poles  for  one  or  more 
branches,  the  functions  P^,  P^,  .  .  .  ,  P„  are  rational  functions 
of  z.     In  this  case  w  is  an  algebraic  function  of  z. 


ALGEBRAIC   FUifCTIONS.  83 


Art.  33.    Algebraic  Functions. 

Any  algebraic  function  satisfies  an  equation  of  the  form 
F(z,  w)  =  o,  where  i^(z,  w)  is  a  rational  entire  function  of  z  and  w. 
It  this  equation  is  of  the  n\h  degree  in  w,  to  any  value  of  z  will 
correspond,  in  general,  n  dififerent  values  of  w,  but  for  special 
values  of  z,  two  or  more  values  of  w  may  be  equal. 

The  principle  of  continuity  appHed  to  the  values  of  an  alge- 
braic function  would  lead  us  to  expect  that,  when  F{a,  'w)  =  o 
has  q  roots  equal  to  h,  it  should  be  possible,  whatever  the  value 
of  the  positive  number  £,  to  determine  a  positive  quantity  d  such 
that,  whenever  \z  —  a\<d,  the  equation  i^(z,  'w)  =  o  would  give  q 
and  only  q  values  of  w  satisfying  the  condition  \w  —  h\<£. 

It  is  necessary  in  the  demonstration  of  this  fundamental  prop- 
erty of  algebraic  functions  to  consider  only  the  case  where  a  and 
h  are  both  zero;  for  every  other  case  can  be  reduced  to  this  one 
by  means  of  the  substitution  z=a+2',  w  =  h-\-'u/.  Write  the 
function  Fiz,  w)  in  the  form 

F{z,  'w)  =  P^  +  P^w-\-, .  .-VPqW^-\-. .  .+P„7t;«, 

in  which,  when  z=o,  P^==P^  =  .  .  .=Pq.^=^o,  but  Pq  takes  a 
value  different  from  zero.  This  expression  can  be  put  in  the 
form 

F{z,w)=P^w%i  +  U^-V\ 
where 

W^Pq^'"^W     Pq' 

Describe  about  the  points  z=o  and  7£;=o  as  centers,  in  the 
is-plane  and  le^-plane  respectively,  circles  C  and  T,  of  radii  r  and 
p.  It  is  possible  to  choose  r  and  p  sufficiently  small  to  satisfy 
the  following  conditions:  (i)  whenever  z  and  w  are  interior  to  C 
and  r, 

\U\<h\ 


84 


FUNCTIONS   OF   A    COMPLEX  VARIABLE. 


(2)  whenever  w  is  on  the  circumference  T,  and  z  is  interior  to  C, 

\V\<i. 

It  is  evidently  possible  to  satisfy  the  first  condition.     The  ine- 
quality 


l^'V, 


+  ...  +  - 
P 


shows,  further,  since  P^, .  .  .  ,  Pg-^  all  approach  zero  with  r, 
that  for  any  value  of  |0,  r  can  be  chosen  sufficiently  small  to  sat- 
isfy the  second  condition. 

But  for  any  assignable  position  of  z  within  C,  the  number  of 
roots  of  the  equation  F{z,  w)  =  o  contained  within  F  is,  by 
Theorem  II,  Article  27,  equal  to 


2mJ    ■ 


dw^ 


[PgW%i  +  U+V)] 


•dw, 


T      PgW%i-\-U-{-V) 

or  the  total  variation  of  any  branch  of 

hglPgW^ii  +  U+V)], 

when  w  describes  the  circumference  T,  divided  by  27Ti.     But 

log  [PgW^{i  +  U+  F)]=log  Pg  +  q  log  w+log  (i  +  U+  V). 

The  first  term  is  constant;  the  total  variation  of  the  second  term 
is  2mq;  and,  since  1C7+  F|  <  i  when  w  is  on  the  circumference  jT, 
the  argument  of  i-\-U-\-V  must  return  to  its  original  value,  and 
the  total  variation  of  log(i  +  C/+F)  is  zero.  The  number  of 
values  of  w  within  F  is,  therefore,  equal  to  q. 

Those  values  of  z  for  which  two  or  more  values  of  w  are  equal 
must  satisfy  the  equation  obtained  by  eliminating  w  between 


F{z,w)=o, 


dw 


F(z,w)=o. 


INTEGRALS  OF  ALGEBEAIC  FUNCTIONS.  85 

For  every  other  finite  value  of  z,  the  equation 

dF(z,  w) 
dw  dz 


dz         '^F{z,  w) 
dw 

gives  at  once  a  single  determinate  value  for  the  derivative  of  w. 

It  follows  from  the  preceding  Article  that  any  branch  Wa  of 
w  must  be  expressible  in  the  neighborhood  of  any  singular  point 
a  A;  by  a  series  of  the  form  * 

Wa  =  A^+A,(z-ak)^  +A^(z-ak^  +. . . 

I  2_ 

-{■B,iz-ak)~^  +  B^(z-ak)   ^+... 

uniformly  convergent  in  a  small  circular  band  surrounding  the 
point  dk.     If  dk  is  not  a  branch  point,  p  =  i. 

Art.  34.    Integrals  of  Algebraic  Functions. 

In  determining  the  value  of  the  integral  of  an  algebraic  func- 
tion w=/{z)  along  any  path  joining  z^  to  z,  it  is  possible  by  virtue 
of  Cauchy's  Theorem  to  alter  the  path  of  integration  arbitrarily, 
provided  that  no  singular  point  is  contained  in  the  region  enclosed 
between  its  original  and  final  positions.  By  employing  the  same 
reasoning  as  in  Article  28,  any  path  joining  z^  to  z  may  be  reduced 
to  a  determinate  path,  preceded  by  a  system  of  loops,  of  which 
each  encloses  a  single  singular  point.  The  value  of  the  integral 
corresponding  to  a  loop  surrounding  a  branch  point  requires 
special  consideration.  If  z  describes  such  a  loop,  w  returns  to 
Zq  with  an  altered  value.  When,  however,  the  initial  point  is 
fixed,  the  value  of  the  integral  is  not  altered  by  varying  arbi- 

*  For  examples  see  Briot  and  Bouquet,  Fonctions  elliptiques  (1875),  PP-  40> 
57;  Chrystal,  Algebra,  vol.  11  (1889),  pp.  356,  370. 


86  FUNCTIONS  OF   A   COMPLEX   VARIABLE. 

trarily  the  form  of  the  loop,  provided  that  no  singular  point  is 
introduced  into  or  removed  from  the  loop. 

To  show  that  a  given  loop,  containing  a  branch  point  and 
attached  to  the  path  of  integration  at  a  point  c^,  different  from 
Zq,  may  be  transformed  into  one  whose  initial  point  is  z^,  it  is 
necessary  to  observe  that  the  variable  passes  first  from  Zq  to  c^ 
and  then  around  the  loop  to  c^  again.  If  now,  before  continuing 
along  the  remaining  part  of  the  path,  z  be  required  to  retrace  its 
way  to  Zq  and  then  return  to  Ci,  the  value  of  the  integral  will  not 
be  altered  thereby;  for  the  integral  resulting  from  the  path  c^z^c^ 
is  equal  to  zero.  The  loop,  however,  has  been  made  to  begin  and 
end  at  z^i  and  it  is  followed  by  a  path  which  begins  at  z^. 

For  any  algebraic  function,  therefore,  just  as  for  a  function 
without  branch  points,  the  most  general  path  of  integration  can 
be  reduced  to  a  determinate  path,  having  the  same  limits,  pre- 
ceded by  a  system  of  loops  of  which  each  encloses  a  single  sin- 
gular point. 

The  integral  around  such  a  loop  enclosing  a  a-,  a  singular 
point  but  not  a  branch  point  for  the  branch  of /(z)  considered,  is 
equal  to  ±27:iBk,  where  Bk  is  the  residue  of  this  branch  of /(z) 
at  a  A,  and  the  plus  or  minus  sign  is  taken  according  as  the  loop 
is  described  in  a  positive  or  negative  direction. 

Consider  now  a  loop  enclosing  a  branch  point  a^.  It  can  be 
reduced  to  a  special  form,  consisting  of  a  small  circle  described 

about  a^^  as  a  center  and  a  line, 
straight  or  curved,  joining  this 
circle  to  z^.  The  term  Q^  to  be 
added  to  the  integral  on  account 
of  this  loop  will  be  obtained  by 
integrating  w=/(z)  from  z^  along 
the  line  joming  z^  to  tnc  circle,  around  the  circle,  and  back  along 
the  same  line  again  to  z^.  The  parts  resulting  from  tracing  the 
line  joining  z^  to  the  circle  in  opposite  directions  do  not  cancel; 
since  on  account  of  the  nature  of  the  branch  point  w  does  not 
take  its  former  system  of  values  when  z  retraces  its  path  to  z^. 

If  now  the  integral  of  /(z)  along  any  determinate  path  from 


INTEGKALS   OF  ALGEBRAIC   FUl^CTIONS.  87 

Zq  to  z  be  denoted  by  I{z),  the  general  value  of  the  integral,  J{z), 
resulting  from  an  arbitrary  path  between  the  same  limits,  is 

J(z)=I(z)  +  2Q„, 

where  Q^  is  the  value  of  the  integral  along  the  wth  loop  in  the 
reduced  form  of  the  path  of  J(z). 

If  the  upper  limit  z  of  the  integral  J(z)  is  situated  in  the  neigh- 
borhood of  a  critical  point,  w  is  expressible  in  a  region  containing 
z  by  the  uniformly  convergent  series 

I  _£ 

w=AQ+A^(z-aky  ^A^iz-aky  +. . . 

'  I  2 

+B,{z-ak)    ^+R,iz-ak)    ^+... 

The  integral,  therefore,  except  for  a  constant  term,  which  includes 
IQ^i  is  equal  to 

J{z)=A^(z-ak)  +  j^A,(z-ak)  ^  +j^A^(z-ak)  ^  +. . . 

+~-^B,(z-ak)~^~-^--^B^(z-ak)~P~  +  . ,  .+pBp.,(z-ak)'^ 

pi  p      2 

_- 1      p  _£_ 

+Bp\og{z-ak)-pBp+z{z-ak)   p --Bp+^{z-ak)  ^-... 
As  an  example  consider  the  integral 


where  the  initial  value  of  the  radical  vi— z^  is  +i.  If  under 
the  integral  sign  z  be  replaced  by  z/,  where  /  is  a  real  quantity  vary- 
ing from  zero  to  unity,  the  resulting  integral 

/(z)  =  z/^-=4== 

will  correspond  to  a  rectilinear  path  joining  the  origin  to  z. 


B8  FUNCTIONS   OF  A   COMPLEX  VARIABLE. 

In  J{z)  the  only  singular  points  of  the  integrand  are  z=  ±1. 
The  integral  for  the  circumference  of  a  small  circle  described 
about  either  of  these  points  as  a  center,  by  Theorem  I  of  Article 
19,  approaches  zero  as  a  limit  simultaneously  with  the  radius  of 
the  circle.  A  loop  enclosing  the  point  +1,  therefore,  gives  a 
term  equal  to 

r'     dz  n^     dz       _      />*     dz     _ 


the  radical  taking  a  negative  sign  on  the  way  back  to  the  origin 
by  virtue  of  the  fact  that  z  has  turned  around  the  branch  point 
2  =  1.  In  the  same  way,  a  loop  enclosing  the  point  2=  —  i  will 
give,  if  the  initial  value  of  the  radical  is  positive, 


dz 


When  z  describes  a  loop  about  either  of  the  points  ±1,  the  radical 
returns  to  the  origin  with  its  sign  changed.  Hence,  if  z  describe 
in  succession  two  loops  about  the  same  branch  point,  the  total 
effect  on  the  value  of  the  integral  is  zero.  If  the  path  of  the  in- 
tegral J^{z)  is  equal  to  that  of  the  integral  J{z)  preceded  by  a 
single  loop  enclosing  the  point  +1  or  the  point  —  i,  the  value 
of  J^{z)  will  be 

Tz—J{z)     or     —Tz—J{z) 

respectively.  If  the  path  of  J^{z)  consist  of  two  loops,  the  first 
about  z  =  I,  the  second  about  z=  —  i,  followed  by  the  path  of  /(2),* 

/i(z)  =  27r+/(z). 

An  arbitrary  path  from  z^  to  z  gives  an  integral  of  the  form 

2mz-\-I{z)     or     (2W  +  i)7r— /(z), 


rUKCTIONS  OF   SEVERAL  VARIABLES.  89 

where  n  is  an  integer  positive  or  negative  and  I{z)  is  the  integral 
for  a  rectilinear  path. 

Prob.  28.  If   i?=V(2— <Zi)  .  . .  (2— a„),    and    the   rectilinear   inte- 
eral  value  of    /     ^  is 


0    i^ 


2m^Ai+  .  .  .  +2w^^^+Z    or    2miAi+  . . .  +2m„A^+AK—Z, 
where  mi, . . .  ,  /^„  are  any  integers,  positive  or  negative. 

Art.  35.    Functions  or  Several  Variables. 

Let  f(Zi,  Z2)  be  a  function  of  two  independent  variables  holo- 
morphic  with  respect  to  each  when  z^  and  z^  are  interior  to  the 
regions  A^  and  A^  respectively.  Let  Q  and  Cg  be  two  closed 
curves  drawn  in  these  regions,  and  let  a^  and  aj  be  points  con- 
tained within  these  curves.     Then 

rAz^z^),         ...      .    ' 

J  - — —dz,  =  2mf(a,,Z2) 

*^Cx   ^1  — "1 

X  1 — —dz^=2mf{a^,  a^), 


so  that 


/fe,  ^2) 


Differentiating  this  integral  with  respect  to  the  parameters  a^  ^j* 
gives  the  general  result 

f*    r         /(Zi,  z^dz^dz^ 


(2;ri) 


'ba^'ba^ 


90  FUNCTIONS  OF  A   COMPLEX    VARIABLE. 

It  follows  that  /(2i,  Zj)  has  an  infinite  number  of  successive  par- 
tial derivatives  holomorphic  under  the  same  conditions  as4tself. 

Let  M  be  the  upper  bound  of  the  modulus  of  /(Zj,  Zj)  when 
Zj  and  Z2  vary  along  the  curves  Q  and  Cj  respectively;  r^  and  rj* 
the  shortest  distances  from  a^  and  a^  to  these  curves;  l^  and  /j, 
the  lengths  of  these  curves:  then 

^1-2  .  .  .  p'i-2  .  .  .  q      MIJ2 

<  (27:)'  r/+^r2«+^* 

If  Cj  and  C2  are  circles  described  about  a^  and  aj  as  centers, 
/i  =  27rri,  l^  =  2nr2,  and 

It  is  easy  now  to  extend  Taylor's  Series  to  the  case  of  a  function 
of  two  variables.  Let/(Zi,  Zj)  be  holomorphic  as  long  as  z^  and 
Z2  remain  within  circles  Cj  and  C^  described  about  a^  and  a^  as 
centers.  Let  flj  +  Zj,  ^2  +  ^2  be  points  chosen  arbitrarily  within 
these  circles.    Then 


I    /    3         3  \^, 
+  7:rV'3^+''3a;M'^''"''^) 

(•-v  Ci    \  3 


DIFFERENTIAL  EQUATIONS.  91 

The  proof  that  the  remainder  approaches  zero  as  a  limit  is  anal- 
ogous to  that  given  in  the  case  of  a  single  variable. 

Corresponding  results  can  be  obtained  for  functions  having 
any  nimiber  of  independent  variables. 

Art.  36.    Differential  Equations.* 
Consider  the  differential  equation 
dw 

where  /(z,  7£;)is  holomorphic  when  z  and  w  are  near  the  points 

Zq  and  Wq  respectively.      By  the  transformation  'W='WQ-h'u/j  z= 

dii/ 
z^\z\  the  equation  becomes  -jy  =  ^{z',  v/),  where   9^(2',  w)  is 

holomorphic  when  2'  and  w'  are  both  near  zero.  Without  loss 
of  generahty,  therefore,  the  discussion  can  be  restricted  to  the 
special  case  where  /(z,  w)  is  holomorphic  with  respect  to  z  and 
w^  when  z  and  w  are  confined  to  small  regions  containing  z  =  o 
and  w  =  o  respectively. 

If  the  given  differential  equation  admits  an  integral,  holomor- 
phic in  the  neighborhood  of  z  =  o,  and  vanishing  at  that  point, 
this  integral  will  be  unique;  for  all  its  successive  differential  co- 
efficients at  the  point  z  =  o  can  be  obtained  from  the  given  differ- 
ential equation.  It  is  sufficient  to  differentiate  that  equation 
once,  and  make  z  =  o,  w=o,  in  order  to  find  the  second  differen- 
tial coefficient;  to  differentiate  again  and  make  the  same  substi- 
tution to  find  the  third  differential  coefficient,  and  so  on.  In  this 
way  is  obtained  the  development. 

'dw^ 
dz 


«'=(:^)/+r^(S)„^'+- •  •=«.^+''.^'+- •  ■ 


If  this  development  can  be  proved  to  converge  when  \z\  is  suffi- 
ciently small,  w  thus  defined  satisfies  the  differential  equation. 

div 
For  -7-  and/(z,  w)  have  the  same  value  for  z=o;  and  their  suc- 

*  Briot  and  Bouquet,  Fonctions  EUiptiques,  p.  325 ;  Picard,  Traite  d' Analyse, 
vol.  n,  p.  291. 


92  FUNCTIONS   OF   A    COMPLEX    VARIABLE. 

cessive  differential  coefficients  with  respect  to  z  of  any  order  what- 

dw 
soever  are  also  equal  for  2=0.    Hence  -7-  and/(z,  w)  are  equal. 

Describe  small  circles  C  and  C  about  the  points  z  =  o,  w=o 
as  centers  with  radii  r,  /.  Let  M  be  the  upper  bound  of  the 
modulus  of  /(z,  w)  within  or  upon  these  circles.  If  now  the 
function 

F(z,w)  = 


Hi'-?) 


be  constructed,  it  will  be  holomorphic  within  the  circles  C  and 
C.  Its  development  in  a  convergent  series  of  ascending  powers 
of  z  and  w,  is  found  by  multiplying  together  the  series  for 


and 


z  w 

r  r 

and  introducing  into  each  term  the  constant  factor  M, 

The  successive  partial  derivatives  of  F{z,  w)  are  all  positive 
and  such  that 

?>zpdw^  \izl<\   'dzp'dw<i  y^rg- 

Consider  now  the  differential  equation 

dW     ^,    „,, 

If  it  has  an  integral  W^  holomorphic  in  the  neighborhood  of 
2=0,  the  integral  will  be  expressible  in  the  form 

The  coefficients  in  this  series  are  all  positive,  and  for  every  value 
of  w 


DIFFERKXTIAL   EQUATIONS.  93 

The  series  given  above  for  w,  therefore,  is  convergent  at  every 
point  where  the  series  for  W  converges.  But  it  is  easy  to  demon- 
strate the  existence  of  the  function  W.     For  the  equation 

dW  M 

dz 


(-7)(-f) 


may  be  written  in  the  form 


/  _W\dW      M 
V     r'l  dz~       z 


I  — 
r 


The  two  members  are  the  derivatives  respectively  of 

Mr  log  [i-^. 


W-~    and 
2r 


If  the  logarithm  be  chosen  so  that  it  vanishes  when  z  =  o,  it  will 
be  holomorphic  within  the  circle  |z|  =  r.  Since  W  is  to  vanish 
when  z=o,  the  relation  between  W  and  z  should  be 


or 


where  the  radical  is  equal  to  + 1  for  z  =  o. 

The    function     W    thus     defined     satisfies     the     equation 

dW 

~-j~=F{z,  W);    it  vanishes  when  z  =  o;    and  it  is  holomorphic 

in  the  interior  of  a  circle  having  for  its  center  the  origin,  and  for 
its  radius  p  the  root  of  the  equation 


2Mr        I      p\ 
i  +  -^log(^i--j=o, 


94  FUNCTIONS   OF   A   COMPLEX  VARIABLE. 

that  is,  p=r(i-e''^Mr\ 

The  series  for  W,  consequently,  converges  in  the  interior  of  the 
circle  of  radius  p.  The  series  for  w  must  converge  in  the  same 
circle.  Hence  the  given  differential  equation  admits  an  integral 
vanishing  for  z  =  o,  and  holomorphic  within  the  circle  of  radius 
p  and  center  at  the  origin. 

The  preceding  discussion  can  be  extended  without  modifica- 
tion to  the  case  of  n  equations : 


-3r=/„(2,^i,^2,...,wJ, 


The  functions  in  the  second  members  are  supposed  to  be  holo- 
morphic with  respect  to  z,  Wj, . . .  ,  w^  within  a  circle  of  radius 
r  described  about  z=o,  and  circles  of  radius  /  described  about 
w;i=o,  ...,w„  =  o. 

If,  further,  M  denotes  the  upper  bound  of  the  moduli  of/, 
fii'-'ifn  i^  ^^^  regions  considered,  the  associated  differential 
equations  are 

dz         dz     '"       dz      ^^^'^i^^^y-'y^nh 

where 

M 


F(z,W,,W,,..,,W)  = 


(-;-)(-^^)-(.-^-)- 


DIFFERENTIAL    EQUATIONS.  95 

The  functions  W^,  PTj, .  .  .  ,  TF„  all  vanish  for  z=o  and  are 
identical,  so  that  only  one  equation 

dW  M 

dz 


(-r)(-7) 


need  be  considered.    The  radius  p  of  the  circles,  within  which 
all  the  developments  converge,  is 


,=r\i~e  («+^)^'-). 


P- 

As  an  example  take  the  differential  equation 
dw 

assuming  as  initial  conditions  2  =  0,  110  =  0.  This  equation  defines 
'Z£;  as  a  holomorphic  function  of  z  in  any  region  in  which  w  re- 
mains finite.  Suppose  that  w  becomes  infinite  for  some  finite 
value  a  of  the  variable  z.  To  determine  the  nature  of  the  point 
2  =  a,  make  the  substitution 

vf 
The  given  differential  equation  is  transformed  to 

dw' 

the  initial  conditions  being  2'  =  o,  ii/  =  o.  This  equation  defines 
i£/  as  a  holomorphic  function  of  z'  in  the  neighborhood  of  2'  =  o, 
and,  consequently,  of  z  in  the  neighborhood  of  z  =  a.  The  given 
differential  equation  is  satisfied,  therefore,  by  a  function  w  whose 
only  finite  critical  points  are  poles. 


96  FUNCTIONS   OF  A   COMPLEX   VARIABLE. 

The  values  of  2  for  which  w  takes  an  assigned  value  may  be 
found  bv  means  of  the  integral 

/""  dw       I     /'"'  dw      j_    n""  dw 
„   i+w^    21  Jq   w  —  i    2i^o   w-^-i 

If  w  describes  two  paths  symmetrical  with  respect  to  the  origin, 
z  acquires  values  numerically  equal  but  of  opposite  signs.  It 
follows  that  w  is  an  odd  function  of  z.  A  loop  enclosing  the  point 
'W=ij  described  in  the  positive  direction  n  times,  adds  to  the  inte- 
gral a  term  equal  to  nn.  A  loop  described  about  w^  —i  in  a 
positive  direction  n  times  similarly  gives  —mz.  The  function  w 
is  thus  periodic,  having  a  period  equal  to  n. 

It  is  possible  to  express  w  as  the  ratio  of  two  functions  having 
no  finite  critical  points.  Assume  'W  =  'wjw2.  The  given  differ- 
ential equation  takes  the  form 


(dw,        \         Idw^        \ 


This  equation  can  be  satisfied  by  making 


dw.  dw^ 


and  2=0,  'Z£'i=o,  ^2  =  1  ^^^7  ^^  taken  as  initial  conditions.     From 
these  equations  can  be  obtained 


w^- 

dw^ 
dz 

d'w, 
dz' 

d'w, 
dz' 

dHv, 
dz' 

w^- 

dw, 
dz 

d'w^ 
dz'~ 

d'w, 
dz'' 

d'w^ 
~  dz* 

Hence, 

when 

z  = 

■Oj 

,    ,  /dw,\  /d'w,\  /d?w\  (d*w,\ 


DIFFERENTIAL     EQUATIONS.  97 

and 

The  series  for  w^  and  w^  are,  therefore, 


Wi  =  -- 1 . . .  =sin0, 

'     I     I -2  -3     I -2 -3 •4-5 

^2=1- -\ :      — ...=cos2: 

1-2  I  -2  -3  -4 

sin  z 

whence  w  = =  tan  z. 

cos  z 

Prob.  29.  Show  that  the  integral  of  —  =  w,  with  the  initial  condi- 
tions  z=o,  10=1,  is  the  series  7£'=i4-zH h. .  .  =  exp.  z. 

Prob.  30.  Show  that  the  equation  ;7-^+«^=o  is  equivalent  to  the 

du  dv  ,    ,  .,,..., 

system,  ■j-='^,  -i-=—u\    and  that  with  the  initial  conditions  z=o, 

u=a,  v=b,  the  solution  by  series  gives  u=a  cos  z+b  sin  0. 

Prob.  31.  Show  that  the  equation  -ty=(i  — w2)(i  — ifeW^  with  the 

az 

initial  conditions  z=o,  w=o,  Vi  — w^=  +  i,  Vi  — )^2'Z£;2=  +  i,  is  equiv- 
alent to  the  system  ~=uv,  j-=—v'w,  —=—k^wu,  with  th^   initial 

conditions  z=o,  w=o,  u=i,  v=i,  and  that  the  functions  w,  u,  v  have 
no  finite  critical  points  except  poles.  The  functions  are  Jacobi's 
■elliptic  functions  sn  z,  en  z,  dn  z,  respectively. 


INDEX. 


Absolute  convergence,  3 
Algebraic  functions,  83 
Analytical  continuation,  80 
Argtiment,  2 

Bound,  33 
Branch  point,  27 

Cauchy's  theorem,  37 
Complex  integrals,  36 
Complex  variable,  2 
Conformal  representation,   11 
Conjugate  functions,  20 
Continuation,  Analytical,  80 
Continuity  of  function,  5 
Continuity  of  variable,  2 
Convergence,  Absolute,  3 
Convergence,  Uniform,  49 
Curvilinear  integrals,  42 
Cut,  26 

Derivative,  8 
Differential  equations,  91 

Elementary  functions,  4 
Element  of  function,  80 
Equipotential  curves,  23 
Essential  singular  point,  28 

Fluid  motion,  21 
Fourier's  series,  48 
Function,  Definition  of,  i 
Functions  having  n  values,  81 
Functions  of  several  variables,  89 

Oamma  function,  71 
Goursat's  demonstration,  10 
Graphical  representation,  7 

Holomorphic  function,  31 

Independent  variable,  i 
Infinity,  Point  at,  31 


Integral,  Definition  of,  32 

In  egral  of  algebraic  function,  85 

Integral  of  uniform  function,  63 

Lacunary  functions,  80 
Laurent's  series,  46 

Magnification,  12 
Mercator's  projection,  20 
Meromorphic  function,  31 
Mittag-LeflSer's  theorem,  71 
Modulus,  2 
Monogenic  function,  10 

Non-monogenic  function,  10 

Ordinary  point,  30 
Orthogonal  systems,  21 

Path,  2 

Pole,  26 

Power  series,  54 

Primary  factors,  69 

Rectifiable  curve,  32 
Residues,  61 

Series  of  functions,  49 
Singular  lines  and  regions,  79 
Singular  points,  25 
Sink,  25 

Spherical  representation,  19 
Stereographic  projection,  20 
Stream  lines,  22 
Synectic  function,  31 

Taylor's  series,  44 

Uniform  convergence,  49 

Vortex  point,  24 

Weierstrassian  elliptic  functions,  71 
Weierstrass's  theorem,  66 

99 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

RENEWALS  ONLY— TEL.  NO.  642^405 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 

Renewed  books  are  subject  to  immediate  recall. 


^pR20W'U«i& 

Rec. 

APR  1 3  1970 

^?.V.t?oT.«.3.                       vJSg^r... 

<;fl  V 


Mm 


